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IN  MEMORIAM 
FLORIAN  CAJORl 


EIGHTH  GRADE 
MATHEMATICS 


By 

Harry  M.  Keal 

Head  of  the  Mathematics  Department 

Cass  Technical  High  School 

Detroit,  Michigan 

and 


Nancy  S.  Phelps 

Grade  Principal 

Southeastern  High  School 

Detroit,  Michigan 


1 


nfi^nius 


ATKINSON,  MENTZER  ^  COMPANY 

NEW  YORK  CHICAGO  ATLANTA  DALLAS 


^ 


COPYRIGHT,  1917,  BY' 
ATKINSON,  MENTZER  &  COMPANY 


Introduc 


tion     I 


THE  growth  of  this  series  of  Mathematics  for  Secondary  Schools, 
has  covered  a  period  of  seven  years,  and  has  been  simultaneous 
with  the  growth  and  development  of  the  shop,  laboratory,  and 
drawing  courses  in  Cass  Technical  High  day  school,  as  well  as  in  the 
evening  and  continuation  classes. 

The  authors  have  had  clearly  in  mind  the  necessity  of  first  developing 
a  sequence  of  mathematics  that  would  enable  the  student  to  recognize 
fundamental  principles  and  apply  them  in  the  shop,  drawing  room,  and 
laboratory;  and,  second  to  so  develop  the  course  that  each  year's  work 
would  be  a  unit  and  not  depend  upon  subsequent  development  for 
intelligent  application. 

It  has  been  assumed  that  the  school  work-shop,  drawing  room,  and 
laboratory  would  furnish  opportunity  to  apply  mathematics  and  that  it 
was  not  necessary  to  exhaust  every  possible  application  in  the 
mathematics  class. 

The  authors  have  been  aware  of  the  popular  demand  for  a  closer  union 
of  algebra  and  geometry,  but  have  recognized  that  demand  only  when 
the  union  came  about  naturally  and  would  assist  the  mathematical 
sequence  desired. 

Instructors  in  the  wood  shop,  pattern  shops,  machine  shop,  drawing 
rooms,  chernistry,  physics,  and  electrical  laboratories,  etc.,  have  furnished 
examples  of  mathematical  apph cation  incident  to  the  respective 
subjects.  Hundreds  of  problems  arising  in  the  industries,  have  been 
brought  in  by  the  machinists,  sheet  metal  workers,  carpenters,  electrical 
workers,  pattern  makers,  draughtsmen,  etc.,  etc.,  coming  to  the  evening 
and  continuation  classes.  Complete  charts  of  machine  shop  work  and 
electrical  distribution  requirements  have  been  made,  including  a 
statement  of  the  required  sequence  of  mathematics.  All  of  this  material 
has  been  classified,  with  a  view  to  the  mathematical  sequence. 

The  net  result  is  a  series  of  Mathematics  so  organized  that  a  mastery 
of  the  text  makes  it  possible  for  a  student  to  use  mathematics  intelli- 
gently in  the  various  departments  of  the  school,  in  the  industries,  and 
at  the  same  time  prepare  for  college  mathematics. 

E.  G.  ALLEN, 

Director  Mechanical  Department, 
Cass  Technical  High  School, 
Detroit,  Mich. 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/eighthgradematheOOkealrich 


TABLE  OF  CONTENTS 


PAGE 

CHAPTER  I 
The  Equation 1 


CHAPTER  II 
Evaluation 15 

CHAPTER  III 
The  Equation  Applied  to  Angles 25 

CHAPTER  IV 

Algebraic    Addition,    Subtraction,    Multiplica- 
tion AND  Division 38 

CHAPTER  V 
Ratio,  Proportion  and  Variation 79 

CHAPTER  VI 
Pulleys,  Gears  and  Speed 96 

CHAPTER  VII 
Squares  and  Square  Roots 107 

CHAPTER  VIII 

Formulas  123 

V 


CHAPTER  I 
THE  EQUATION 


% 


10 


^ 


Fig.  1 


^ 


1  In  order  to  find  the  weight  of  an  object,  it  was  placed  on 
one  pan  of  perfectly  balanced  scales  (Fig.  1).  It,  together 
with  a  3-lb.  weight,  balanced  a  10-lb.  weight  on  the  other  pan. 
If  3  lbs.  could  be  taken  from  each  pan,  the  object  would  be 
balanced  by  7  lbs.  This  may  be  expressed  by  the  equation, 
x+3  =  10,  in  which  the  expressions  x+3  and  10  denote  the 
weights  in  the  pans,  the  sign  ( = )  of  equality  denotes  the  per- 
fect balance  of  the  scales,  arid  x  is  to  be  found. 

2  Equation:  An  equation  is  a  statement  that  two  expres- 
sions are  equal.  The  two  expressions  are  the  members  of  the 
equation,  the  one  at  the  left  of  the  equality  sign  being  called 
the  first  member,  and  the  one  at  the  right,  the  second  member. 

3  From  the  explanatory  problem,  it  will  be  seen  that  the 
same  number  may  be  subtracted  from  both  members  of  an  equation. 

Oral  Problems: 
Solve  f or  X :  • 

1.  x+7  =  21  3.     x+1. 1=3.5 

2.  x+2  =  3  4.     x+2|=7^ 

1 


5.     x+f  = 


5  _  il 
12 


THE  EQUATION 


Fig.  2 


Jj.  It  is  required  to  find  the  weight  of  a  casting.  It  is  found 
that  3  of  them  exactly  balance  a  10-lb.  weight  (Fig.  2).  If  the 
weight  in  each  pan  could  be  divided  by  3,  one  casting  would  be 
balanced  by  3j  lbs.     This  may  be  expressed  by  the  equation, 

3x=10, 

X  =  3i.  (dividing  both  members  by  3) 

5  From  this  explanatory  problem,  it  will  be  seen  that  hoik 
members  of  an  equation  may  he  divided  hy  the  same  number. 

Oral  Problems: 
Solve  for  x : 

1.  4x  =  12 

2.  2x=16 

3.  5x  =  9 

4.  llx  =  33 
6.  l.lx=12.1 

Example :    Solve  f or  x :    5x + 1 2  =  37 

5x  =  25    Why? 
x=   5    Why? 


THE  EQUATION 

Exercise 

1 

Solve  for  the  unknown: 

1. 

x+l  =  5 

11. 

9x+8=116 

2. 

x+7  =  9 

12. 

7w+5f  =  12f 

3. 

2a+6  =  16 

13. 

28t+14  =  158 

4. 

3x+7  =  28 

14. 

3x+4j  =  9 

5. 

5s+17  =  62 

16. 

15s+.5  =  26 

6. 

9x+12  =  93 

16. 

llx+J=8_9 

7. 

2x+l  =  6 

17. 

1.2x+2  =  14 

8. 

5y+3  =  15 

18. 

4.6x+8  =  100 

9. 

4n+3.2  =  15.2 

19. 

6.3x+2.4=15 

10. 

12m+8  =  98 

20. 

7.1m+.55  =  9.07 

Q 


A 


ATA 


fxibrlfsn 


10 


Fig.  3 


^ 


6  If  an  apparatus  is  arranged  as  in  Fig.  3,  it  is  seen  that  if 
the  upward  pull  of  2  lbs.  be  removed,  2  lbs.  would  have  to  be 


4  THE  EQUATION 

put  upon  the  other  pan  to  keep  the  scales  balanced.     This 
may  be  expressed  by  the  equation, 

4x-2=10 

4x  =  12   (Adding  2  to  both  members) 
X  =  3      (Dividing  both  members  by  4) 

7  From  this  problem,  it  will  be  seen  that  the  same  number 
may  he  added  to  both  members  of  an  equation. 

Oral  Problems: 


Solve  for  x: 

1. 

3i-4  =  8 

2. 

7x-l  =  15 

3. 

4x-f  =  7} 

4. 

5x-.l  =  .9 

6. 

2x-i  =  6i 

Exercise  2 

Solve  for  the  unknown : 

1. 

x-7  =  10 

11. 

13r-21=44 

2. 

2x-13  =  ll 

12. 

12s-35  =  41 

3. 

5x-17  =  13 

13. 

7f-4  =  26 

4. 

4x-ll  =  25 

14. 

4x-3  =  16 

6. 

3x-7=15 

15. 

9x-3.2=14.8 

6. 

12x-4  =  44 

16. 

3m-2=3.1 

7. 

7m-5  =  31 

17. 

14x-5  =  21 

8. 

4x-18=18 

18. 

2.1x-3.2  =  3.1 

9. 

17t-3i=13f 

19. 

.5y-4  =  5.5 

10. 

llx-9  =  90 

20. 

3x-9j  =  8.5 

SIMILAR 

TERMS 

Exercise  3. 

Review 

Solve  for  the  unknown : 

1.    9x-8=46 

6.    3w-lJ  =  lf 

2.    8x-7  =  53 

7.     19t-.2  =  3.6 

3.     5x+7  =  28 

8.    6.37n+3.92  =  73.99 

4.     28m-9  =  251 

9.     .4x+.02=.076 

5.     16y+13  =  73 

10.     2s+2i  =  9f 

11.     Two  times  a  number  increased  by  43  equals  63.     Fi 

the  number. 

12.  If  10  be  added  to  3  times  a  number,  the  result  is  50. 
What  is  the  number? 

13.  Five  times  a  number  decreased  by  6  equals  39.     Find 
the  number. 

14.  If  55  be  subtracted  from  7  times  a  number,  the  result 
is  22.     What  is  the  number? 

15.  If  to  57  I  add  twice  a  certain  number,  the  result  is 
171.    What  is  the  number? 

18.     State  the  first  five  problems  in  this  exercise  in  words. 

How  many  yards  of  cloth  are  7  yds.  and  5  yds.? 
How  many  dozens  of  eggs  are  12  doz.  and  3  doz.? 
How  many  bushels  of  wheat  are  8  bushels  and  1 1  bushels? 
How  many  b's  are  4  b's  and  7j  b's? 
How  many  x's  are  3x  and  9x? 

In  such  expressions  as  2a -|-3x+4+2x+7+3a,  2a  and  3a 
may  be  combined,  3x  and  2x,  and  also  4  and  7,  making 
the  expression  equal  to  5a+5x+ll.  2a  and  3a,  3x  and 
2x,  4  and  7  are  called  similar  terms. 


6  THE  EQUATION 

Example  1.     Solve  for  x: 

4x+13x— 7x  =  40 

lOx  =  40     (combining  similar  terms) 
x  =  4.    Why? 
Example  2.     Solve  for  x: 

14x+7-2x  =  43 

12x+7  =  43  Why? 

12x  =  36  Why? 

x  =  3     Why? 

9    8-7+3  =  ?  8x-7x+3x  =  ? 

8+3-7  =  ?  Similarly   8x+3x-7x  =  ? 
3-7+8  =  ?  3x-7x+8x  =  ? 

10  These  problems  illustrate  the  principle  that  the  value  of 
an  expression  is  unchanged  if  the  order  of  its  terms  is  changed, 
provided  each  term  carries  with  it  the  sign  at  its  left. 

NOTE:    If  no  sign  is  expressed  at  the  left  of  the  first  term,  the  sign 
(+)  is  understood. 

Example  1.     15-3x+llx  =  39 

8x+15  =  39  Why? 

8x  =  24  Why? 

x  =  3  Why? 

Example  2:      lly-4+21  =  50 

lly+17  =  50  Why? 

lly  =  33  Why? 

y  =  3  Why? 


ORDER   OF  TERMS  / 

Exercise  4 

Solve: 

1.  4x-x  =  12 

2.  llx+3x  =  35 

3.  14x-3x=44 

4.  3x+7x=90 

5.  9y-9y+8y  =  40 

6.  4s+3s-2s  =  17 

7.  3.2x+2.3x  =  110 

8.  1.3y-2.7y+3.3y  =  57 

9.  11.2x+7.8x  =  57 

10.     l.ls-1.4s+lls  =  26.75 

Exercise  5 
Solve: 

1.  x-18  =  17  9.  12x-8x+6+3x  =  8+12 

2.  x+18  =  21  10.  25x+20-7x-5+5x  =  56+5 

3.  2y-16  =  30  11.  8x+60+4x-50+3x-7x  =  20 

4.  3m-m  =  21  12.  2-2x+7x=42.5 
6.  3m-l  =  23  13.  3yH-1.2+2y=46 

6.  6.5x-l.l  =  50.9  14.  x-1.25x+12.7+3.5x  =  38.7 

7.  4x+3x-3  =  25  16.  2x+ 15.8 -2.3x+14.5x  =  186.2 

8.  lly-4y-7  =  28  16.  6.15y-1.65y+7.8  =  57.3 

17.  8y+6.875+2y=46.875 

18.  z-8.73+5.37z  =  61.34 

19.  5t-8.75t+6.87+8t  =  57.87 

20.  3.73x-9.23+15x  =  65.69 


8  THE  EQUATION 

11  Equations  often  arise  in  which  the  unknown  appears  in 
both  members.  In  that  case,  aim  to  make  the  term  containing 
the  unknown  disappear  from  one  member,  and  the  one  contain- 
ing the  knowrij  from  the  other  member. 

Example  1:    3x  — l=x+3 

3x  =  xH-4    (adding  1  to  both  members). 
2x  =  4     (subtracting  X  from  both  members). 
X  =  2     Why? 

Note  that  in  adding  or  subtracting  a  term  from  both  members,  it  must 
be  combined  with  a  similar  term. 

Example  2: 

5x+4-3x-l  =  7-x+2 

2x4-3  =  9  — X  (combining  similar  terms  in  each  member). 
2x  =  6-x    Why? 
3x  =  6     Why? 
x  =  2     Why? 


Exercise  6 


Solve: 

1.     2x-6  =  x 


2.  2x+3  =  x+5 

3.  13x-40  =  8+x 

4.  7y-7  =  3y+21 
6.  9x-8  =  25-2x 

6.  20+10x  =  38+4x 

7.  3x+9+2x+6  =  18+4x 

8.  5x+3-x  =  x+18 

9.  7m-18+3m=12+2m+2 

10.  18+6m-f-30+6m  =  4m-|-8H-12+3m+3+mH-29 


CLEARING   OF   FRACTIONS  9 

11.  25x+20-7x-5  =  56-5x+5 

12.  10x-61-12x+27x  =  8x-41+20+4x+25 

13.  25f+5x+6x+9|-2x=180-8x-8| 

14.  2.8x+39.33+x  =  180-1.2xH-32.09-7.16 

15.  5x+26f+9x  =  360-5x-143f 

12  If  an  object  in  one  pan  of  scales  will  balance  a  4-lb.  weight 
in  the  other,  it  will  be  readily  seen  that  5  objects  of  the  same 
kind  would  need  20  lbs.  to  balance  them.  This  may  be 
expressed  by  the  equation,  x  =  4 

5x  =  20  (multiplying  both  members  by  5). 

13  From  this  problem,  it  will  be  seen  that  both  members  of  an 
equation  may  be  multiplied  by  the  same  number. 

This  principle  is  needed  when  the  equation  contains  frac- 
tions. The  process  of  making  fractions  disappear  from  an 
equation  is  called  clearing  of  fractions. 

tJf.  RULE:  To  clear  an  equation  of  fractions,  multiply  both  members 
by  the  lowest  common  denominator  (L.  C.  D.)  of  all  the  fractions 
contained  in  the  equation. 


Example  1 


Example  2:    -— ==  — 


^»= 

=  4 

x+6  = 

=  8  (multiplying  both  members  by  2). 

x  = 

=  2     Why?      . 

r     r  _ 
3     Y 

_16 
'  3 

7r-3r  = 

=  112 

(multiplying  both  members  by  21). 

4r  = 

=  112 

Why? 

r  = 

=  28 

Why? 

10  THE  EQUATION 

„  ,^      m_3,m7m 

Example  3:    —  — 3t7t+  — =  -  — — 
4        1^      5      5      3 

15m- 198+ 12m  =  84 -20m  Why? 

27m -198  =  84 -20m  Why? 

47m -198  =  84  Why? 

47m  =  282  Why? 

m  =  6  Why? 

15  The  four  principles  used  thus  far  may  be  more  generally 
stated  as  follows : 

1.  //  equals  are  added  to  equals,  the  results  are  equal. 

2.  //  equals  are  subtracted  from  equals,  the  results  are  equal, 

3.  //  equals  are  multiplied  by  equals,  the  results  are  equal. 

4 .  //  equxds  are  divided  by  equals,  the  results  are  equal. 


Solve: 


Exercise  7 

1. 

5^-^  =  10 
3      6 

6. 

x_2_x 
2    3     6 

2. 

?+?  =  9 
5    4 

7. 

y  =  ?+16 
3     7 

3. 

?i'+?r=i7 

3      4 

8. 

?x+3  =  ^+4 

4. 

x+Jx  =  6 

9. 

2x     x_x      1 
9      6     18    3 

6. 

x-|x  =  7 

10. 

3x     1      X  ,_ 
7"3  =  27+' 

PRINCIPLES    OF   EQUATIONS  11 

11.     l|s+fs  =  s+13  13.     -+4r--=26+lJr 

^7  4  3 


^         ^^  4  2     3     4     10 

15.     7x+y+-+23=-+5ix+113 

16  Sometimes  it  is  convenient  to  make  the  term  containing 
the  unknown  disappear  from  the  first  member,  and  the  one 
containing  the  known,  from  the  second. 


Example 

1: 

x+6  = 

=  3x- 

-2 

6  = 

=  2x- 

-2 

Why? 

8  = 

=  2x 

Why? 

x  = 

=  4 

Why? 

Example 

2: 

L^  =  4 
3x 

16=  12x     Why?  (L.  C.  D.  is  3x) 
x  =  l|       Why? 

Exercise  8 


Solve: 

1. 

L5  =  5 

a 

2. 

5=15 

a 

3. 

1=2 
4x 

,      3x     7 

4.     —  =  - 

4      2 


^      16     2x 
6.      — =  — 

5       3 


6.  14  =  x+9 

7.  17  =  2x-3 


12  THE  EQUATION 

8      x+10  =  2x-9  12.     ^+47=-+4n 

2  7 


9.     2x-2i  =  5x-17}  13.     ^-l=lZ?-^-2ia 

^  ^  2  3       3 


10.  7x+20-3x  =  60+4x-50+8x    14.     .lx+6.2  =  .3x+.2 

11.  3m+60  =  15m-f3-2m+7  15.     10H-.lx  =  5+|x 


Exercise  9 

Solve: 

1.    7m_8  =  5i-—  5.     2_t     5_t^t     t     ^^ 

6  12  3      9     6    2 


2.     7x-8  =  6x+ix  6.     1+''—'^  =  '^-^ 

^  2     5     6     3     4 


3.     ??-^=25i-^-  7.     y-^+2i  =  -^-?:+^ 

56  3  12    8484 


4.     ??+3  =  ^x-2  8.     -x-^x+4|  =  3x+- 

3  6  3      5         ^  15 

9.     lly^x-l^x-302  =  60+l|x+183 


10.     x-3|+Jx  =  9j-^ 


PROBLEMS  13 

Exercise  10 

1.  Five  times  a  certain  number  equals  155.  What  is  the 
number? 

2.  Four  times  a  number  increased  by  7  equals  43.  Find  the 
number. 

3.  Twelve  times  a  number  decreased  by  .18  is  equal  to 
17.82.     Find  the  number. 

4.  There  are  three  numbers  whose  sum  is  72.  The  second 
number  is  three  times  the  first,  and  the  third  is  four  times  the 
first.     What  are  the  numbers? 

5.  The  sum  of  two  numbers  is  12  and  the  first  is  4  more 
than  the  second.    What  are  the  numbers? 

6.  If  10  is  subtracted  from  three  times  a  number,  the 
result  is  twice  the  number.     Find  the  number. 

7.  If  1^  of  a  number  is  increased  by  6,  the  result  is  30. 
Find  the  number. 

8.  The  sum  of  J,  ^  and  ^  of  a  number  is  26.  What  is  the 
number? 

9.  Divide  19  into  two  parts  so  that  one  part  is  5  more 
than  the  other. 

10.  Divide  19  into  two  parts  so  that  one  part  is  5  times 
the  other. 

11.  Divide  $24  between  two  persons  so  that  one  shall 
receive  $2j  more  than  the  other. 

12.  A  farmer  has  4  times  as  many  sheep  as  his  neighbor. 
After  selling  14,  he  has  3^  times  as  many.  How  many  had 
each  before  the  sale? 


14  THE  EQUATION 

13.  Two  men  divide  $2123  between  them  so  that  one  receives 
$8  more  than  4  times  as  much  as  the  other.  How  much  does 
each  receive? 

14.  Three  candidates  received  in  all  1020  votes.  The  first 
received  143  more  than  the  third,  and  the  second  49  more  than 
the  third.     How  many  votes  did  each  receive? 

16.  A  man  spent  a  certain  sum  of  money  for  rent,  f  as 
much  for  groceries,  $2  more  for  coal  than  for  rent,  and  $28 
for  incidentals.  In  all  he  paid  out  $100.00.  How  much  did 
he  spend  for  each? 

16.  A  farmer  has  24  acres  more  than  one  neighbor  and  62 
acres  less  than  another.  The  three  together  own  one  square 
mile  of  land.     How  much  has  each? 

17.  A  man  traveled  a  certain  number  of  miles  on  Monday, 
•f-  as  many  on  Tuesday,  f  as  many  on  Wednesday  as  on  Mon- 
day, and  on  Thursday  10  miles  less  than  twice  as  many  as  he 
did  on  Monday.  How  far  did  he  travel  each  day  if  his  trip 
covered  82  miles? 

18.  One  man  has  3  times  as  many  acres  of  land  as  another. 
After  the  first  sold  60  acres  to  the  second,  he  had  40  acres 
more  than  the  second  then  had.  How  many  acres  did  each 
have  before  the  transaction? 

19.  One  boy  has  $10.40  and  his  brother  has  $64.80.  The 
first  saves  20  cents  each  day,  and  his  brother  spends  20  cents 
each  day.     In  how  many  days  will  they  have  the  same  amount? 

20.  A  man  after  buying  27  sheep  finds  that  he  has  1^ 
times  his  original  flock.     How  many  sheep  had  he  at  first? 


CHAPTER  II 

EVALUATION 

17  Definite  Numbers:  The  numerals  used  in  arithmetic  have 
definite  meanings.  For  example,  the  numeral  7  is  used  to 
represent  a  definite  thing.  It  may  be  7  yards,  7  pounds,  7 
cubic  feet  or  7  of  any  other  unit.  Also  in  finding  the  circum- 
ference of  a  circle,  we  multiply  the  diameter  by  w  which  has  a 
fixed  value.  Numerals  and  letters  which  represent  fixed  values 
are  called  definite  numbers. 

18  General  Numbers:  The  area  of  a  rectangle  is  found  by 
multiplying  the  base  by  the  altitude.  This  may  be  expressed 
by  bXa,  in  which  the  value  of  b  may  be  10  ft.,  6  in.,  30  rds., 
or  any  number  of  any  unit  used  to  measure  length,  and  a  may 
be  any  number  of  a  like  unit.  Letters  which  may  represent 
different  values  in  different  problems  are  called  general  numbers. 

19  Signs:  When  the  multiplication  of  two  or  more  factors  is 
to  be  indicated,  the  sign  of  multiplication  is  often  omitted  or 
expressed  by  the  sign  (•)•  Thus  7XaXbXm  is  written 
7-a-b-m  or  more  often  7abm. 

NOTE:    Care  should  be  taken  in  the  use  of  the  sign  (•)  to  distinguish 
it  from  the  decimal  point.     7-9  means  7X9,  7.9  means  7i^o. 

W  Coefficient:  The  expression  7abm  may  be  thought  of  as 
7ab-m,  7  abm,  7-abm,  or  7b -am,  etc.  7ab,  7a,  7,  and  7b 
are  called  the  coefficients  of  m,  bm,  abm,  and  am  respectively. 

1.  In  the  following,  what  are  the  coefficients  of  x*^    4abx; 
^xyz;  17mxw. 

2.  Name  the  coefficients  of  ab  in  the  following:     S^axby; 
fmabz;  .Obnsa. 

3.  What  is  the  coefficient  of  17  in  17mxw? 

15 


16  EVALUATION 

The  coefficient  of  a  factor  or  of  the  product  of  any  number 
of  factors,  is  the  product  of  all  the  remaining  factors.  In 
8axy,  8  is  the  numerical  coefficient.  The  numerical  coefficient 
1  is  n£ver  written,     laxy  is  written  axy. 

21  Power:  If  all  the  factors  in  a  product  are  the  same,  as 
x-x-x-x,  the  product  is  called  a  power,  x-x-x-x  is  read  ''x 
fourth  power"  and  is  written  x*-  a- a- a- a- a  is  read  "a  fifth 
power"  and  is  written  a^.  b-b  or  b^  is  *'b  second  power"  but 
is  more  often  read  ^'b  square."  In  the  same  way  b-b-b  (b^) 
is  called  "b-cube." 

22  Exponent:  The  small  number  written  at  the  right  and 
above  a  number  is  called  its  exponent  and  it  indicates  the  power 
of  the  number.  The  exponent  1  is  never  written,  x  means 
x^  or  ''x  first  power." 

23  Base:  The  number  to  be  raised  to  a  power  is  called  the 
base. 

Name  the  numerical  coefficients,  bases  and  exponents  in  the 
following : 

V^x^,  Sjaio,  3.7m2n^  f^r^  l|m  l|m^ 

24  Sign  of  Grouping:  The  Sign  of  Grouping  most  commonly 
used  is  the  parenthesis  (  )  and  means  that  the  parts  enclosed 
are  to  be  taken  as  a  single  quantity.  For  example,  3(x-y) 
means  that  x-y  is  to  be  multiplied  by  3  making  3x  -  3y.  (x  -y)^ 
means  (x-y) (x-y) (x-y). 

25  Evaluation:  Evaluation  of  an  expression  is  the  process  of 
finding  its  valu^  by  substituting  definite  numbers  for  general 
numbers  in  the  expression,  and  performing  the  operations 
indicated. 


EVALUATION   OF   EXPRESSIONS  17 


Example  1 :    Evaluate  4:ix^x^  if  a  =  3,  x  =  2. 
4a2x3  =  4.32.2'3=4-9-8  =  288. 


a2        Sb*    ,    m^ 

Example  2:    Find  the  value  of  ~, ;   ~^  TTt 

^                                           m^         c2         2a3 

when  a=l,  b  = 

=  2,  c  =  5,  m  =  2. 

a2      5b4       m^ 
m^       c2       2a3 

P        5-24          25 

2^          5^      '    2V 

1         80        32 
~    8         25         2 

=i-?-« 

5-128+640 

40 

40         ^^^^ 

Example  3:    Evaluate  a(a  — b+y^)  when  a  =13,  b  =  3,  y  =  4. 
a(a-b+y2)  =  13(13 -3+42) 

=  13-26 

=  338 

Exercise  11 

Evaluate  the  following  if  a  =  8,  b  =  6,  c  =  4,  d  =  2,  x  =  9: 

1.  2x  7.  3x2 

2.  x2  8.  (3x)2 

3.  3x  9.  llax 

4.  x3  10.  2abcd 
6.    4x  11.  2a2x3 
6.    x^  12.  x2-a2 


18  EVALUATION 

13.  x(a+b)  17.    —:-+:; 

X    b     d 

14.  4b(x-c)  18.     (x+a)(c-d) 

15.  a2+2ab+b2  19.      Viad 

16.  c2-2cd+d2  20.     ab(c-3) 


Exercise  12 

Find  the  value  of  the  following,  when  a  =  2,  b  =  3,  c  =  7, 
d  =  4,  m=l,  x  =  5: 

1.    iaVc  11.     (3x+7)(c-2) 


2.  x3-a3  12.      Vb2+d2 

3.  x'+d^  3a2 

13.     —  (x2-c2+25) 

4.  3b2-4m2  bd 

6.  xM-a^m  14.    a3(x-c+3m)(c2+d2) 
e!     2a2x3(c-d)  15.     ^^ 

7.  4+^  16-     ^(x2+a2-b2)(c2-d2-m2) 
a^       d  2d 

8.  ^(x+a)c  17.      Vx(a+b) 

9.  ^a3x2c(b3-d2)  18.     ^d(a+b)+c 
c2    x2 


^^-     {:^  +  l^— 7  ^^-     ^5x(a+b) 

b^      d^    a^ 

20.     (a+b)(b+c)-(b+c)(x+d)  +  (x+d)(d+m) 


PERIMETER   FORMULAS 


19 


Evaluation  of  Formulas 

26  A  Formula  is  the  statement  of  a  rule  or  principle  in  terms 
of  general  numbers.  For  example,  distance  traveled  is  equal  to 
rate  times  time. 


Formula,  d  =  r-t 


Iwt 


Example  1 :    Evaluate  b  =  —  (Formula  for  board  feet) 

whenl  =  16',  w  =  8",  t  =  2" 
168. 2 


b  = 


12 


=  21 J 


Example  2:  Evaluate  A  =  ^h(b+bO  (Area  of  a  trapezoid) 
ifh  =  3A,b=12i  b'  =  6i 

A  =  i.3^(12H6i) 

A  _  1    q  3     .18^ 

1  51  75 

2*16'  4 

3825 

128 


A= 


29yV8  or  29.102- 


Perimeter  Formulas 

27  The  perimeter  of  a  figure  enclosed  by  straight  lines  is  the 
sum  of  its  sides. 

6 


a 


Fig.  4.    Square 


Fig.  5.     Rectangle 


20 


EVALUATION 


Fig.  6.     Triangle 


Fig.  7.    Quadrilateral 


Exercise  13 


1.  The  perimeter  of  a  square  (Fig.  4)  is  equal  to  4  times 
one  side.     P  =  4a.     Find  P,  if  a  =  9. 

2.  Find  the  value  of  P,  in  P  =  4a,  if  a  =  ij. 

3.  Find  the  value  of  P,  in  P  =  4a,  if  a  =  1.175. 

4.  The  perimeter  of  a  rectangle  (Fig.  5)  is  equal  to 
a+b+a+b  =  2a+2b  =  2(a+b).  P  =  2(a+b).  Find  P,  if 
a  =  3,  b  =  5. 

5.  Find  P,  in  P  =  2(a+b),  if  a  =  |,  b  =  |. 

6.  Find  P,  in  P  =  2(a+b),  if  a=  1.7862,  b  =  2.1324. 

7.  The  perimeter  of  a  triangle  (Fig.  6)  is  expressed  by  the 
formula,  P  =  a+b+c.     Find  P,  if  a  =  7,  b  =  ll,  c=19. 

8.  Evaluate  P  =  a+b+c,  if  a  =  f,  b  =  f,  c  =  f. 

9.  Find  the  value  of  P,  in  P  =  a+b+c,  if  a  =  7.621,  b  = 
8.37,  c  =  1.3. 


PERIMETER    PROBLEMS  21 

10.  The  perimeter  of  a  quadrilateral  (Fig.  7)  is  expressed 
by  the  formula,  P  =  a+b+c+d.  Find  P,  if  a  =  20,  b  =  15, 
c=13,  d=14. 

11.  Evaluate  P  =  a+b+c+d,  when  a=lf,  b  =  lf,  c  =  1y^, 
d  =  li. 

12.  Find  P,  in  P  =  a-f-b+c+d,  if  a  =  172.32,  b  =  96.3, 
c  =  81.04,  d  =  56.2. 


Exercise  14.     Equations  Involving  Perimeters 

1.  The  perimeter  of  a  square  is  96.     Find  a  side. 

2.  The  perimeter  of  a  triangle  is  114.  The  first  side  is  6 
less  than  the  second  and  24  less  than  the  third.      Find  the  sides. 

3.  Find  the  dimensions  of  a  rectangle  whose  perimeter  is 
48  if  the  length  is  3  times  the  width. 

4.  Find  the  dimensions  of  a  rectangle  if  its  length  is  4 
more  than  the  width  and  xts  perimeter  is  82. 

6.  The  length  of  a  rectangle  is  4  more  than  twice  the 
width  and  its  perimeter  is  ISS^^.     Find  the  length. 

6.  The  perimeter  of  a  rectangle  is  48.648.  Find  the  width 
if  it  is  J  of  the  length. 

7.  The  perimeter  of  a  rectangle  is  94.  The  width  is  11.3 
more  than  ^  of  the  length.     Find  the  length  and  the  width. 

8.  The  perimeter  of  a  quadrilateral  is  176.  The  first  side 
is  J  of  the  second,  the  third  is  8  more  than  the  second,  and  the 
fourth  is  3  times  the  first.     Find  the  sides. 


22 


EVALUATION 

Exercise  15.    Area  Formulas 


Fig.  8.     Rectangle 


Fig.  9.     Parallelogram 


b 

Fig.  10.     Triangle 


I 


Fig.  11.    Trapezoid 


1.  The  area  of  a  rectangle  (Fig.  8)  is  equal  to  the  base 
multiplied  by  the  altitude.  A  =  a-b.  Find  A,  if  a  =11.5, 
b=18.6. 

2.  Evaluate  A  =  a.b,  if  a  =  2|,  b  =  3f. 

3.  Express  the  result  of  problem  2  in  decimal  form. 

4.  The  area  of  a  parallelogram  (Fig.  9)  is  the  base  times 
the  altitude.     A  =  a-b.     Find  A,  if  a=ly^g^,  b  =  6.71. 

6.  The  area  of  a  triangle  (Fig.  10)  is  \  the  product  of  the 
base  and  altitude.     A  =  ib.h.     Find  A,  if  b  =  12.23,  h.  =  6.57. 

6.  Evaluate  A  =  ^b.h,  if  b  =  9f,  h  =  4|. 

7.  The  area  of  a  trapezoid  (Fig.  11)  is  J  the  product  of 
the  altitude  and  the  sum  of  the  parallel  sides.  A  =  Jh(b+b'). 
Find  A,  if  h  =  10f,  b  =  19f,  b'  =  12f 

8.  Express  the  result  of  problem  7  in  a  decimal  correct 
to  .001. 


CIRCLE   AND   GENERAL   FORMULAS  23 

Exercise  16.    Circle  and  Circular  Ring  Formulas 


Fig.  12.     Circle 


Fig.  13.    Circular  Ring 


1.  C  =  27rT.    (Fig.  12).     Find  C,  if  ;r  =  3.1416  (See  art.  17) 
r=li 

2.  C  =  ttB.     Find  C,  if  D  =  5.724. 

3.  A  =  7rr\     Find  A,  if  r=  l|. 

4.  A=.7854D2.     Find  A,  if  D  =  5.724. 

5.  A  =  ;r(R2-r2)  (Fig.  13).     Find  A,  if  R  =  7i,  r  =  4j. 


Exercise  17.     General  Formulas 

Evaluate  the  following  formulas  for  the  values  given: 

1.  P  =  awh,  if  a  =  120,  w  =  .32,  h  =  9|. 

2.  W=-.p,  if  1  =  25,  h  =  4j,  p  =  60. 

h 

3.  F  =  ljd+iifd  =  lf. 

4.  L=lfd+|,  if  d  =  2i. 

6.     S  =  2gt^,  if  t  =  4.    (g  is  a  definite  number.     Its  value  is  32.16), 

6.  S  =  |gt^+vt,  if  t  =  3,  V  =  7. 

7.  D=  Va2+b2+c2,  if_a  =  3,  b  =  4,  c=12. 

8.  V  =  |h(b'+b+  Vb-bO,  if  h  =  2f,  b=12,  b'  =  3. 
uv 


9.     F  = 


u+v, 
10.     V  =  |^r3,  if  r  =  2.3. 


if  u  =  11.5,  v  =  6.5. 


24  EVALUATION 

Checking  Equations 

28  The  solution  of  an  equation  may  be  tested  by  evaluating 
its  members  for  the  value  of  the  unknown  quantity  found. 
If  its  members  reduce  to  the  same  number,  the  value  of  the 
unknown  is  correct. 

Example:     2x+?^5^^^  =  3x+l. 
5 

_    ,  6x-2     _     ,  , 

2xH ^-  =  3x4-1.       Why? 

o 

10x+6x-2=  15x+5.     Why? 
x  =  7.  Why? 

Check: 

5 

14+8  =  21  +  1. 
22  =  22. 

Exercise  18 


Solve  and  check: 

1.  6y-7  =  3y+20.  ^      2(x+2) 

2.  ll=3x+9.  ^ 


=  7. 


X  V    ^  8.    Z(?±^.^  =  f+2. 

3.     3-1  =  2-2.  12         6    4 


4.     2(2x+5)  =  13. 


9.     2x-l  =  f(5-x)-l|. 

7(5-x) 


6. 


6(z-6)  =  z+8.  N^*^-    i(5-x)  = 


6.     ?^^  =  3.  10.     ?(x+l)+^^-l=4i 

5  5  4       5 


CHAPTER  III 
THE  EQUATION  APPLIED  TO  ANGLES 


29  Angle:  If  the  line  OA  (Fig.  14) 
revolves  about  O  as  a  center  to  the 
position  OB,  the  two  lines  meeting  at 
the  point  O  form  the  angle  AOB.  The 
point  O  is  called  the  vertex  of  the  angle 
and  the  lines  OA  and  OB  are  called 
the  sides  of  the  angle. 


Fig.  14.     Angle 


Fig.  15.    Right  Angle 


B  A 

Fig.  16.     Straight  Angle 


0  A 

Fig.  17.    Perigon 


30  Right  Angle:  If  the  line  turns 
through  one  fourth  of  a  complete  rev- 
olution (Fig.  15),  the  angle  is  called  a 
Right  Angle. 


31  Straight  Angle:  If  the  line  turns 
through  one  half  of  a  complete  rev- 
olution (Fig.  16),  the  angle  is  called  a 
Straight  Angle. 

32  Perigon:  If  the  line  turns  through 
a  complete  revolution  (Fig.  17),  re- 
turning to  its  original  position,  the 
a-ngle  is  called  a  Perigon, 


How  many  right  angles  in  a  straight  angle? 
How  many  right  angles  in  a  perigon? 
How  many  straight  angles  in  a  perigon? 


25 


26 


THE   EQUATION 


Fig.  18.    Protractor 

S3  A  Protractor  (Fig.  18)  is  an  instrument  used  for  measuring 
and  constructing  angles.  On  it,  a  straight  angle  is  divided 
into  180  equal  parts  called  degrees,  written  180°. 

How  many  degrees  in  a  right  angle? 

How  many  degrees  in  a  perigon? 


Fig.  19 

Drawing  Angles 
34  Example:    Draw  an  angle  of  37°. 

Using  the  straight  edge  of  the  protractor,  draw  a  straight 
line  OA.  Place  the  straight  edge  of  the  protractor  along  the 
line  OA,  with  the  center  point  at  O.      Count  37°  from  the  point 


THE   PROTRACTOR 


27 


where  the  curved  edge  touches  OA  and  mark  the  point  B 
(Fig.  19).  Again  use  the  straight  edge  of  the  protractor  to 
connect  the  points  O  and  B. 


Exercise  19 


1.  Draw  an  angle  of  30°. 

2.  Draw  an  angle  of  45°. 

3.  Draw  an  angle  of  60°. 

4.  Draw  an  angle  of  120°. 

5.  Draw  an  angle  of  135°. 


6.  Draw  an  angle  of  150°. 

•    7.  Draw  an  angle  of  18°. 

8.  Draw  an  angle  of  79°. 

9.  Draw  an  angle  of  126°. 
10.  Draw  an  angle  of  163°. 


Measuring  Angles 


Fig.  20 

35  Example:     Measure  the  angle  AOB. 

Place  the  straight  edge  of  the  protractor  along  one  side  of 
the  angle  as  OA,  with  its  center  at  the  vertex  of  the  angle 
(Fig.  20).  Count  the  number  of  degrees  from  the  point  where 
the  curved  edge  of  the  protractor  touches  OA  to  the  point 
where  it  crosses  the  line  OB.     The  angle  AOB  contains  54°. 


28 


THE  EQUATION 


F/GZl 


F/G  28 


Exercise  20 


1.  Measure  the  angle  in  Fig.  21. 

2.  Measure  the  angle  in  Fig.  22. 

3.  Measure  the  angle  in  Fig.  23. 

4.  Measure  the  angle  in  Fig.  24. 


6.  Measure  the  angle  in  Fig.  26. 

7.  Measure  the  angle  in  Fig.  27. 

8.  Measure  the  angle  in  Fig.  28. 

9.  Measure  the  angle  in  Fig.  29. 


6.  Measure  the  angle  in  Fig.  25.      10.  Measure  the  angle  in  Fig.  30 


Reading  Angles 


36  Reading  Angles:  An  angle  is 
read  with  the  letter  at  the  vertex 
between  the  two  letters  at  the 
ends  of  the  sides.  The  angle  1 
in  Fig.  31  is  read  BAG  or  CAB 
and  is  written  Z  BAG  or  Z  GAB. 
ReadtheangleZ2;Z3.    (Fig.31). 


Fig.  31 


READ  NG   ANGLES 


29 


A    r/G34  S 


A        r/G  35 


Exercise  21 


1.  Read  the  Zs  1,  2,  3,  (Fig.  32). 

2.  Read  the  Zs  1,  2,  3,  4,  (Fig.  33). 

3.  Read  the  Zs  1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  (Fig.  34). 

4.  Read  the  Zs  1,  2,  3,  4,  5,  (Fig.  35). 


30  THE  EQUATION 

Exercise  22 

1.  Measure  the  ZCAD  (Fig.  31). 

2.  Measure  the  Z  ACB  (Fig.  32). 
.  3.  Measure  the  ZCDA  (Fig.  33). 

4.  Measure  the  ZEFA  (Fig.  34). 

5.  Measure  the  ZBGF  (Fig.  35). 


Fig.  36 


37  Zl  +Z2  +  Z3  +Z4  =  ZAOB 
(Fig.  36).  If  AGE  is  a  straight  Hne, 
the  Z  AOB  contains  180°.     Therefore 

Zl+Z2+Z3+Z4  =  180°. 

^     38  The  sum  of  all  the  angles  about  a 
point  on  one  side  of  a  straight  line  is  180°. 


Fig.  37 


Exercise  23 


Fig.  38 


Fig.  39 


1.  Find  X  in  Fig.  37.     Check  with  a  protractor. 

2.  Find  x  in  Fig.  38.     Check. 

3.  Find  the  unknown  angle  in  Fig.  39.     Check. 

4.  Three  of  the  four  angles  about  a  point  on  one  side  of 


a  straight  line  are  16°,  78°,  51°,  respectively, 
angle. 


Find  the  fourth 


ANGLE    EQUATIONS  31 

5.  Find  the  three  angles  about  a  point  on  one  side  of  a 
straight  line  if  the  first  is  twice  the  second,  and  the  third  is 
three  times  the  first. 

6.  Draw  with  a  protractor  the  angles  of  problem  5  a^  in 
Figs.  37,  38,  39. 

7.  Find  the  three  angles  about  a  point  on  one  side  of  a 
straight  line  if  the  first  is  twice  the  third,  and  the  second  is 
a  right  angle. 

8.  Draw  the  angles  of  problem  7. 

9.  Find  the  four  angles  about  a  point  on  one  side  of  a 
straight  line  if  the  second  is  5°  less  than  the  first,  the  third  is 
6°  more  than  the  first,  and  the  fourth  is  68°. 

10.     Draw  the  angles  of  problem  9. 

Exercise  24 

Example :    . 

The  three  angles  about  a  point  on  one  side  of  a  straight  line 

4  X 

are  represented  by  x+6°,  ^x  — 12°,  and  78°  —  ^.      Find  x  and 

the  angles. 

x+6+|x-12+78-|=180°.       Why? 

o  o 

3x+18+4x-36+234-x  =  540.         Why? 
6x+216  =  540.         Why? 
6x  =  324.         Why? 
x  =  54.  Why? 

x+6  =  54+6  =  60°      1st  angle. 
|x  - 12  =  72  - 12  =  60°    2nd  angle. 

78  - 1  =  78  -  18  =  60°    3rd  angle. 
o 

NOTE:    The  fact  that  the  sum  of  the  angles  found  is  180°  checks  the 
problem. 


32  THE  EQUATION 

If  tne  angles  about  a  point  on  one  side  of  a  line  are  repre- 
sented by  the  following,  find  x  and  the  angles: 

1.  |x,  x+4,  lix+2. 

2.  fx-2,  iVx-f  7,  3(x+7)  Jx+19. 

3.  4(x+l),  7(2x-ll),  127-6X. 

4.  3x-i,  2x,  2f  (2x+l),  |(x+6). 
6.  ix+40,  2x-9,  129.18-2X. 

6.  Find  the  angles  about  a  point  on  one  side  of  a  straight 
line  if  the  first  is  25°  more  than  the  second,  and  the  third  is 
three  times  the  first. 

7.  Find  the  angles  about  a  point  on  one  side  of  a  straight 
line  if  the  first  is  6  times  the  second,  plus  16°,  and  the  third 
is  J  of  the  first,  minus  4°. 

8.  Find  the  five  angles  about  a  point  on  one  side  of  a 
straight  line  if  the  second  is  J  of  the  first,  the  third  is  5°  more 
than  f  of  the  first,  the  fourth  is  10°  less  than  twice  the  first, 
and  the  fifth  is  22^°. 


Fig.  40 


ANGLES  ABOUT  A   POINT  33 

39  Z1+  Z2+  Z6=180°  Why? 

Z7+Z4+Z5  =  180°  Why? 

Therefore,  Z1+ Z2+ Z3+ Z4+ Z5  =  360°. 

40  The  sum  of  all  the  angles  about  a  point  is  360°. 

Exercise  25 

If  all  the  angles  about  a  point  are  represented  by  the  fol- 
lowing, find  X  and  the  angles: 

1.  |x,88-jx,ljx-13,  4(^+11). 

DO  6 

2.  23+-,  136- -,-+93,    -+17. 

4  5'  3  2 

3.  4(x-5),    ?+5li,  3x+47|. 

4.  i(3x-36),  i(2x+15),  -  +30,  82-|x,  x+48j. 

6 

5.  |x+3.15,  3(x+1.75),  J(x+94.05). 

6.  The  sum  of  four  angles  is  a  perigon.  One  is  18°  more 
than  three  times  the  smallest,  another  is  59°  more  than  the 
smallest,  and  the  last  is  18°  less  than  twice  the  smallest.  Find 
the  four  angles. 

Supplementary  Angles 

41  Supplementary  Angles:  If  the  sum  of  two  angles  is  a 
straight  angle  or  180°,  they  are  called  supplementary  angles. 
Each  is  the  supplement  of  the  other. 

Exercise  26 

1.  What  is  the  supplement  of  16°;  92°;  24°;  13|°;  15lf°? 

2.  x  is  the  supplement  of  80°.     Find  x. 


34  THE  EQUATION 

3.  X  is  the  supplement  of  x+32°.     Find  x  and  its  supple- 
ment. 

4.  2x  — 20°  and  7x+47°  are  supplementary  angles.     Find 
X  and  the  angles. 

6.    One  of  two  supplementary  angles  is  24°  larger  than  the 
other.    Find  them. 

6.  The  difference  between  two  supplementary  angles  is 
98°.     Find  them. 

7.  One  of  two  supplementary  angles  is  4  times  the  other. 
Find  the  angles. 

8.  How  many  degrees  in  an  angle  which  is  the  supplement 
of  3j  times  itself? 

9.  One  of  two  supplementary  angles  is  27°  less  than  3 
times  the  other.     Find  the  angles. 

10.  One  of  two  supplementary  angles  is  y  of  the  sum  of  the 
other  and  63°.     Find  the  angles. 

4^  The  supplement  of  an  unknown  angle  may  be  indicated 
by  180° -X. 

Indicate  the  supplement  of  y°;  d°;  fx°;  |^y°. 

When  a  problem  involves  two  supplementary  angles,  but  is 
such  that  one  is  not  readily  expressed  in  terms  of  the  other,  let 
X  equal  one  angle,  and  180— x  the  other. 

Exercise  27 

1.  1^  of  an  angle,  plus  55°  is  equal  to  ^  of  its  supplement, 
plus  4°.     Find  the  supplementary  angles. 

Let  X  =  one  angle 
180 — X  =  other  angle 
then     fx4-55  =  -|(180-x)+4. 

2.  The  sum  of  double  an  angle  and  12j°  is  equal  to  |  the 
supplement  of  the  angle.     Find  the  supplementary  angles. 


COMPLIMENTARY   ANGLES  35 

3.  If  an  angle  is  trebled,  it  is  30°  more  than  its  supplement. 
Find  the  supplementary  angles. 

4.  If  an  angle  is  added  to  J  its  supplement,  the  result  is 
128°.     Find  the  supplementary  angles. 

5.  If  f  of  an  angle,  minus  16°,  is  added  to  f  of  its  supple- 
ment, plus  72°,  the  result  is  190°.  Find  the  supplementary 
angles. 

Complementary  Angles 

4S  Complementary  Angles:  If  the  sum  of  two  angles  is  a  right 
angle  or  90°,  they  are  called  complementary  angles.  Each  is 
the  complement  of  the  other. 

Exercise  28 

1.  What  is  the  complement  of  82°;  9°;  71°;  10^°;  43|°? 

2.  X  is  the  complement  of  32°.     Find  x. 

3.  X  is  the  complement  of  x+76°.  Find  x  and  its  com- 
plement. 

4.  fx+ 12°,  and  §xH- 10°  are  complementary  angles.  Find 
x  and  the  angles. 

5.  One  of  two  complementary  angles  is  25°  larger  than 
the  other.     Find  them. 

6.  The  difference  between  two  complementary  angles  is 
37f  °.     Find  them. 

7.  One  of  two  complementary  angles  is  three  times  the 
other.     Find  the  angles. 

8.  How  many  degrees  in  an  angle  that  is  the  complement 
of  2|  times  itself? 

9.  One  of  two  complementary  angles  is  7°  more  than  twice 
the  other.     Find  the  angles. 

10.     One  of  two  complementary  angles  is  f  of  the  sum  of 
the  other  and  23°.     Find  the  angles. 


36  THE  EQUATION 

44  The  complement  of  an  unknown  angle  may  be  indicated  by 
90  — X.     Indicate  the  complement  of  y°;  m°;  fx°;  ^y° 

When  a  problem  involves  two  complementary  angles,  but 
is  such  that  one  is  not  readily  expressed  in  terms  of  the  other, 
let  X  equal  one  angle,  and  90  — x  the  other. 

Exercise  29 

1.  The  sum  of  an  angle  and  \  of  its  complement  is  46°. 
Find  the  angle. 

2.  The  complement  of  an  angle  is  equal  to  twice  the  angle 
minus  15°.     Find  the  angle. 

3.  If  20°  is  added  to  five  times  an  angle,  and  20°  sub- 
tracted from  ^  of  the  complement,  the  two  angles  obtained, 
when  added,  will  equal  114°.     Find  the  angle. 

4.  f  of  an  angle  is  equal  to  f  of  its  complement,  minus 
14°.     Find  the  angle. 

5.  f  of  the  complement  of  an  angle,  plus  15°  is  equal  to 
treble  the  angle.     Find  the  angle. 

Exercise  30 

1.  The  sum  of  J,  ^,  and  f  of  a  certain  angle  is  126°.  Find 
the  number  of  degrees  in  the  angle. 

2.  The  supplement  of  an  angle  is  equal  to  four  times  its 
complement.     Find  the  angle,  its  supplement  and  complement. 

3.  The  sum  of  the  supplement  and  complement  of  an 
angle  is  98°  more  than  twice  the  angle.     Find  the  angle. 

4.  The  complement  of  an  angle  is  20°  more  than  \  of  its 
supplement.     Find  the  angle. 

5.  The  sum  of  an  angle,  J  of  the  angle,  its  supplement, 
and  its  complement  is  243°.     Find  the  angle. 


REVIEW   OF   ANGLES  37 

6.  The  complement  of  an  angle  is  equal  to  the  sum  of  the 
angle  and  J  of  its  supplement.     Find  the  angle. 

7.  An  angle  increased  by  ^  of  its  supplement  is  equal  to 
twice  its  complement.     Find  the  angle. 

8.  ^  the  supplement  of  an  angle  is  equal  to  3  times  its 
complement,  plus  20°.     Find  the  angle. 

9.  The  sum  of  treble  an  angle,  f  of  its  complement,  and 
-|  of  its  supplement  is  equal  to  62°  less  than  a  perigon.  Find 
the  angle. 

10.  Y^Y  of  the  complement  of  an  angle  is  equal  to  J  the 
supplement,  plus  3°.     Find  the  angle. 

11.  The  three  angles  about  a  point  on  one  side  of  a  straight 
line  are  such  that  the  second  is  89°  more  than  ^  of  the  sup- 
plement of  the  first,  and  the  third  is  f  of  the  complement  of 
the  first.     Find  the  three  angles. 

12.  The  sum  of  four  angles  is  223°.  The  second  is  twice 
the  first,  the  third  is  ^  the  supplement  of  the  second,  and  the 
fourth  is  the  complement  of  the  first.     Find  the  four  angles. 

13.  There  are  four  angles  about  a  point.  The  second  is 
^  the  first,  the  third  is  the  supplement  of  the  second,  and  the 
fourth  is  the  complement  of  the  second,  plus  30°.  Find  the 
four  angles. 

14.  There  are  five  angles  about  a  point  on  one  side  of  a 
straight  hne.  The  second  is  ^  of  the  first,  the  third  is  ^  the 
supplement  of  the  second,  the  fourth  is  f  the  complement  of 
the  second,  the  fifth  is  10°.     Find  the  five  angles. 

15.  Express  by  an  equation  that  the  supplement  of  an  angle 
is  equal  to  its  complement,  plus  90°. 

Does  41°  for  x  check  the  equation? 

Does  25°?    Does  153°?    What  values  may  x  have? 


CHAPTER  IV 

ALGEBRAIC  ADDITION,  SUBTRACTION, 
MULTIPLICATION  AND  DIVISION 

Positive  and  Negative  Numbers 

45  1.  The  top  of  a  mercury  column  of  a  thermometer  stands 
at  0°.  During  the  next  hour  it  rises  4°,  and  the  next  5°. 
What  does  the  thermometer  read  at  the  end  of  the  second  hour? 

2.  The  top  of  a  mercury  column  stands  at  0°.  During  the 
next  hour  it  falls  4°,  and  the  next,  it  falls  5°.  What  does  it  read 
at  the  end  of  the  second  hour? 

3.  If  the  mercury  stands  at  0°,  rises  4°,  and  then  falls  5°, 
what  does  the  thermometer  read? 

4.  If  the  thermometer  stands  at  0°,  falls  4°,  and  then 
rises  5°,  what  does  the  thermometer  read? 

6.  If  the  mercury  stands  at  0°,  rises  4°,  and  then  falls  4°, 
what  does  the  thermometer  read? 

6.  A  traveler  starts  from  a  point  and  goes  north  17  miles, 
and  then  north  15  miles.  How  far  and  in  which  direction  is  he 
from  the  starting  point? 

7.  A  traveler  starts  from  a  point  and  goes  south  17  miles, 
and  then  south  15  miles.  How  far  and  in  which  direction  is  he 
from  the  starting  point? 

8.  A  traveler  goes  17  miles  south,  and  then  15  miles  north. 
How  far  and  in  which  direction  is  he  from  the  starting  point? 

38 


POSITIVE   AND   NEGATIVE   NUMBERS  39 

9.    A  traveler  goes  17  miles  north,  and  then  15  miles  south. 
How  far  and  in  which  direction  is  he  from  the  starting  point? 

10.  A  traveler  goes  17  miles  south,  and  then  17  miles  north. 
How  far  is  he  from  the  starting  point? 

11.  An  automobile  travels  35  miles  east,  and  then  40  miles 
east.  How  far  and  in  which  direction  is  it  from  the  starting 
point? 

12.  An  automobile  travels  35  miles  west  and  then  40  miles 
west.  How  far  and  in  which  direction  is  it  from  the  starting 
point? 

13.  An  automobile  travels  35  miles  west,  and  then  40  miles 
east.  How  far  and  in  which  direction  is  it  from  the  starting 
point? 

14.  An  automobile  travels  35  miles  east,  and  then  40  miles 
west.  How  far  and  in  which  direction  is  it  from  the  starting 
point? 

16.  An  automobile  goes  35  miles  east,  and  then  35  miles 
west.     How  far  is  it  from  the  starting  point? 

16.  A  boy  starts  to  work  with  no  money.  The  first  day  he 
earns  $.75,  and  the  second  $.50.  How  much  money  has  he  at 
the  end  of  the  second  day? 

17.  A  boy  has  to  forfeit  for  damages  $.75  more  than  his 
wages  the  first  day,  and  $.50  more  the  second  day.  What  is  his 
financial  condition  at  the  end  of  the  second  day? 

18.  A  boy  earns  $.75  the  first  day,  and  forfeits  $.50  the 
second  day.     How  much  money  has  he? 

19.  A  boy  forfeits  $.75  the  first  day,  and  earns  $.50  the 
second.    How  much  money  has  he? 

20.  A  boy  earns  $.75  the  first  day,  and  forfeits  $.75  the 
second.     How  much  money  has  he? 


40  ADDITION 

46  Such  problems  as  these  show  the  necessity  of  making  a 
distinction  between  numbers  of  opposite  nature.  This  can  be 
done  conveniently  by  plus  (+)  and  minus  (  — ).  If  a  number 
representing  a  certain  thing  is  considered  positive  (plus) ,  then 
a  thing  of  the  opposite  nature  must  be  negative  (minus).  Thus, 
if  north  10  miles  is  written  +10,  south  10  miles  must  be  written 
—  10.  If  east  25  feet  is  written  +25,  west  25  feet  must  be 
written  —25. 

47  If  such  numbers  as  these  are  to  be  combined,  their  signs 
must  be  considered.  Thus  a  rise  of  19°  in  temperature  fol- 
lowed by  a  rise  of  9°  may  be  expressed  as  follows:  (+19°)  + 
(+9°)  =  +28°.  A  trip  15  miles  south  followed  by  one  25 
miles  south  may  be  expressed:  (  —  15)  +  (  —  25)  =  —  40.  A 
trip  42  miles  east  followed  by  one  26  miles  west  is  expressed: 
(+42)  +(-26)  =  +16.  A  saving  of  $1.75  followed  by  an 
expenditure  of  $2.00  is  expressed:    (+1.75) +(-2.00)  =  -.25. 

These  four  problems  may  also  be  written: 
1.      19+9  =  28 


2.  -15-25= -40 

or 

3.  42-26  =  16 

4.  1.75-2.00= -.25 


-    .25 

This  combination  of  positive  and  negative  numbers  is  called 
Algebraic  Addition. 


1. 

+  19 
+  9 

+28 

2. 

-15 
-25 
-40 

3. 

+42 
-26 
+  16 

4. 

+  1.75 
-2.00 

ADDITION   OF   SIGNED   NUMBERS  41 

ADDITION 

j^S  RULE:     To  add  two  numbers  with  like  signs,  add  the  numbers 
as  in  arithmetic,  and  give  to  the  result  the  common  sign. 

To  add  two  numbers  with  unlike  signs,  subtract  the  smaller  number 
from  the  larger,  and  give  to  the  result  the  sign  of  the  larger. 

NOTE:  If  no  sign  is  expressed  with  a  term,  +  is  always  understood. 
Care  should  be  taken  not  to  confuse  this  with  the  absence  of  the  sign  of 
multipHcation.     (See  Art.  19.) 


Exercise  31 

Add 

.: 

1. 

+  19, 

+  10 

16. 

-if 

2. 

-19, 

-10 

17. 

-li 

3. 

-19, 

+  10 

18. 

-3i  2i 

4. 

+  19, 

-10 

19. 

6!,  -8| 

5. 

-10, 

+  19 

20. 

-7h  +7f 

6. 

+  10, 

-19 

21. 

13|,  -23f 

7. 

-75, 

+25 

22. 

-llf,8f 

8. 

+38, 

+  19 

23. 

-2.32,  -1.68 

9. 

+  11, 

-26 

24. 

3.47,  5.43 

10. 

+  10, 

-10 

25. 

8.44,  -7.25 

11. 

-40, 

+39 

26. 

8.75,  -11.25 

12. 

-4,  +26 

27. 

5.732,  -4.876 

13. 

i-i 

1 

28. 

-18.777,  -3.333 

14. 

il 

29. 

-173.29,  239.4 

15. 

1 1 

16>    ~ 

-1 

30. 

-208.21,  171.589 

42  ADDITION 

/fd  1.    Add  -19,  -10 

2.  Add  +11,  +26. 

3.  Add  the  results  of  problems  1  and  2. 

How  does  the  result  of  problem  3  compare  with  the  result  if 
—  19,  —10,  +11,  +26,  were  to  be  added  in  one  problem  as 
follows? 

-19-10+11+26=? 

-19+11  -10  +26  =? 

-19+26+11  -10  =? 

+26  -10  -19  +11  =?  (See  Art.  10.) 

50  RULE:    To  add  several  numbers,  add  all  the  positive  numbers  and 
all  the  negative  numbers  separately,  and  combine  the  two  results. 


Exercise  32 

Add: 

1.  +50,  +41,  -23,  -7. 

2.  +47,  -49,  +2,  -35. 

3.  +3,  -40,  -17,  4. 

4.  82,  18,  -100. 

6.  -79,  -21,  -100. 

6.  -119,  +1,  -21,  -14,  +101. 

7.  -2.36,  +4.24,  5.73,  -8.66. 

8.  -3f ,  5f ,  -4yV 

10.    23|,  -19|,  17f, -111,5^. 


ADDITION   OF   SIMILAR  TERMS  43 

61  Term:  A  term  is  an  expression  whose  parts  are  not  sepa- 
rated by  plus  (+)  or  minus  (  — ).  llx^  —  14abxy,  +23f  are 
terms. 

NOTE:  Such  expressions  as  8(x+y),  3(a— b),  etc.,  are  terms  because 
the  parts  enclosed  in  the  parenthesis  are  to  be  treated  as  a  single  quantity. 
(See  Art.  24.) 

52  Similar  Terms:  Similar  or  like  terms  are  those  which  differ 
in  their  numerical  coefficients  only;  as  2x3yz2,  —^x^yz"^. 

53  Only  similar  terms  can  he  combined. 


Exercise  33 
Add: 

1.  -16r,  18r,  8r. 

2.  4.2s,  ~5.7s,  2s. 

3.  7|x,  -4fx,  -2ix,  X. 

4.  2jab,  4|ab,  —  3|ab,  ab. 

5.  24abc,  —  36abc,  lOabc,  +4abc,  —  abc. 

6.  -32a2b,  40a2b,  -Qa^b,  2a2b. 

7.  3vV,  vV,  -9v2y3,  -4v2y^. 

8.  -3|xVz,  5fxVz,  -4yVxVz. 

9.  3.16xy2z5,  -4.08xy2z5,  QmxyHK 

10.  8(x-y),  -6(x-y),  +4(x-y). 

11.  -12(x+y),  -7(x+y),  -(x+y). 

12.  -6|(c-d),3f(c-d),4|(c-d). 

13.  -8(x2+y2),  24(x2+y2),  17(x2+y2),  +(x2+y2). 

14.  8(x+y+z),  14(x+y+z),  -2(x+y+z). 

15.  Il(x2+y)^  -5(x2+y)4,  24(x2+y)^ 


44  ADDITION 

54  Monomial:  An  expression  containing  one  term  only  is 
called  a  monomial. 

55  Polynomial:  An  expression  containing  more  than  one  term 
is  called  a  polynomial.  A  polynomial  of  two  terms  is  called  a 
binomial,  and  one  of  </iree  terms  a  trinomial. 


Addition  of  Polynomials 

56  Example :    Add  2a3  -  2a2b  -  b^,  -  Tab^  - 1  la^, 
and  b3+7a3+3ab2+2a2b. 

Since  only  similar  terms  can  be  combined,  it  is  convenient 
to  arrange  the  polynomials,  one  underneath  the  other  with 
similar  terms  in  the  same  vertical  column,  and  add  each  column 
separately  as  follows: 

2a3-2a2b-b3 
-lla^  -7ab2 

+7a^+2a^b+b^+3ab^ 
-2a3  -4ab2 


Exercise  34 

Add: 

1.  4a+3b-5c,  -2a-m+3c,  2m-9c+2b,  5a+3m-4b. 

2.  pq+3qr+4rs,  —  pq+4rs— 3qr,  st— 4rs. 

3.  2ax2+3ay2-4z2,  ax2+7ay2-4z2,  2z2+ay2-a2x. 

4.  fa2-|ab-Jb2,  2b2-a2-fab,  -ab-Sb^+fa^. 

5.  3|m-4|x+2jf,  2iVx-f+2jm. 

6.  8.75d-3.125r,  2.873r+7.625f-10d,  4.29f-r+1.25d. 


ADDITION   OF   POLYNOMIALS  45 

7.  3(x+y)-7(x-y),  5(x+y)+5(x-y),   -2(x+y)- 

3(x-y). 

8.  f(a2-b2)-f(b2-c2)+f(c2-a2),   f(a2-b2)-|(c2-a2), 

4(b2_c2)_2i(a2-b2). 

9.  5(x+y)-7(x2+y2)+8(x3+y3),  -4(x'+y')+5{x'^+y')- 

4(x+y),  2(x2+y2)-4(x3+y3)-(x+y). 

10.    6(ab+c)+7(a-m)+a2bc3m,    Sa^bc^m-SCab+c)- 
5(a— m),  3(ab+c)  — (a  — m)— 4a2bc^m. 


SUBTRACTION 

57  1.  If  a  man  is  five  miles  north  (+5)  of  a  certain  point, 
and  another  is  12  miles  north  (+12)  of  the  same  point,  what 
is  the  difference  between  their  positions  (distance  between 
them),  and  in  what  direction  is  the  second  from  the  first? 

2.  If  the  first  man  is  5  miles  south  (  —  5)  of  a  point,  and 
the  second  12  miles  south  (  —  12),  what  is  the  difference  between 
their  positions,  and  in  what  direction  is  the  second  from  the 
first? 

3.  The  first  man  is  at  (  —  5),  and  the  second  is  at  (+12). 
What  is  the  difference  between  their  positions,  and  in  what 
direction  is  the  second  from  the  first? 

4.  The  first  is  at  (+5),  and  the  second  at  (  —  12).  What 
is  the  difference  between  their  positions,  and  in  what  direction 
is  the  second  from  the  first? 

58  To  find  the  difference  between  the  positions  of  the  men  in 
the  above  problem,  the  signs  of  their  positions  must  be  con- 
sidered. Finding  the  difference  between  such  numbers  is 
called  Algebraic  Subtraction, 


46 


SUBTRACTION 


Find  the  difference  between  the  positions  and  the  direction 
of  the  second  man  from  the  first  in  each  of  the  following: 

Second  man 
First  man 


+  12 
+  5 

-12 
-  5 

+  12 
-  5 

-12 

+  5 

+  5 

+  12 

-  5 
-12 

+  5 
-12 

-  5 

+  12 

lowing: 

+  12 
-  5 

-12 

+  5 

+  12 
+  5 

-12 
-  5 

+  5 
-12 

-  5 

+  12 

+  5 

+  12 

-  5 
-12 

How  do  the  results  of  the  corresponding  problems  in  the  two 
groups  compare? 

59  RULE:     To  subtract  one  nximber  from  another,  change  the  sign  of 
the  subtrahend  mentally  and  add. 


Exercise  35 

Subtract: 

1.     +27 

+  12 

4. 

-32 

+21 

7. 

+  16 
-42 

10.      -llf 
+  15i 

2.     -13 

-  8 

5. 
6. 

+  15 

+82 

-  81 
-127 

8. 
9. 

-  39 
+  100 

12| 
-4i 

11.     -fax 
-fax 

3.     +21 
-  5 

12.    7ibV 
6fbV 

13. 

by2 
Ifby^ 

16. 

3a-b+c 
4a       —  c 

14. 

-5  (a+y) 

-li(a+y) 

17. 

3ix2+5fy3-  z 
2ix2-2|y3-2z 

15. 

14.92(m2 
149.2  (m2 

18. 

-a^- 

a^b+ab^ 
a^b          -b^ 

SUBTRACTION   OF   POLYNOMIALS  47 

19.  .3(x+y)-4.8(x2+y2) 

-5.7(x+y)+4.8(x^+y^) 

20.     -5  (ab+c)-10.7(x+y+z)+51a2bxVz+19ab2xy3z. 
-3|(ab+c)  +  l.Q7(x+y+z)-17a^bxVz+20abV^z. 


60  A  problem  in  subtraction  is  often  written  in  the  form 
(  —  19)  — (+7).  In  that  case  it  is  better  to  actually  change  the 
sign  of  the  subtrahend  and  then  the  problem  is  one  of  addition 
instead  of  subtraction  and  is  written: 

(-19)  +  (-7)or  -19-7= -26. 


Exercise  36 

Subtract: 

1.  (-28)-(-36)  6.     (8.91abc)-(-3fabc) 

2.  (+35)-(-2li)  6.     (-2yVVz)-(3.1416x2yz) 

3.  (-|)-(+f)  7.     (3a+2b)-(2a+3b) 

4.  (+2jmx)-(-f5mx)     8.     (5x2-7y2)-(-2x2+y2) 
9.     (-9|m3n+3|mn3)  -  (4f  m^n-Sfmn^) 

10.  (3ja+2b)-(3ia+7c) 

11.  (-1.7a2b2-2.9b4)-(-3.3ab3-4.16b4) 

12.  (4x-5§y+3fz2)-(2ix+6.25y-5jz2) 

13.  (-  11.23a2b2+4jbV-  Ijc2a2)  -  (-  11.22a2b2+ 

4jbV-1.875c2a2) 

14.  (6x2-9mx~  15m2)  -  (Qx^-  IGm^) 

15.  (-a^b^+a^b^+ab^)  -  (a^b-a^b^+b^) 


48  ADDITION   AND   SUBTRACTION 

Exercise  37.     (Review) 

1.  From  the  sum  of  x2-2hx+h2  and  x^-Ghx+Qh^,  sub- 
tract 3x2+ 2hx-4h2. 

2.  Simplify  (x2-2xy+y2)  +  (2x2-3xy+y2)-(3x2-5xy+ 
2y^). 

3.  From  6x^  — 7x— 4,  subtract   the    sum    of   Qx^  — 8x+x' 
and  5  — x^+x. 

4.  Simphfy  (im-in+fp)-(|m-fn+|p)-f  (- iVm- 
|n-|p). 

6.     Subtract  the  sum  of  6— 4x^  — x  and  5x  — 1  —  2x2,  fj-om 
the  sum  of  2x3+7-4x-5x2  and  3x2-6x3-2+8x. 

6.  Find  the  difference  between  (12x4+6x5-2)  +  (6x4- 
8x+14-8x3),  and  (0)-(- 10x3+2- 15x2+llx5-4x). 

7.  Subtract:    3x2-5xy+2y2-2x+  7y 

2x2-  xy+8y2-9x-14y 

8.  Simphfy:     (5a3-2a2b+4ab2)  +  (-9a2b+7ab2+8b3)  + 
(-8a3-ab2+2b3). 

9.  Add:     fgt2  +v-    Jt 

gt2         -|v+    6t 
-1.3gt2  +  llt+.02 

-10v+1.2t+lJ 
6gt2+ 11.625V  -2.25 

Solve  and  check: 

10.  12a+l-(3a-4)  =  2a+8+(4a+4). 

11.  (3x-4)-6=(x-l)-(2x-3). 

12.  -25-(-5+2p)  =  (13p-50). 

13.  (3x-15)-(2x-8)=0 


ADDITION  AND  SUBTRACTION  49 


14.  2k-(— -^)=--(--4|) 

3      6      2       2        ^ 

15.  ix+(?x-?)-(^x+l)  =  lf. 
3        5      5        6       3        ^ 


Signs  of  Grouping 

61  Removal  of  Parentheses:  By  Art.  47,  parentheses  connected 
by  plus  signs  may  be  used  to  express  a  problem  in  addition, 
and  the  parentheses  can  be  removed  without  affecting  the 
signs.     For  example: 

(3a-2b)  +  (2a-3b)  =  3a-2bH-2a-3b-5a-5b. 

By  Art.  62,  two  parentheses  connected  by  a  minus  sign  may 
be  used  to  express  a  problem  in  subtraction,  and  the  paren- 
theses can  be  removed  by  actually  changing  the  sign  of  each 
term  enclosed  in  the  parenthesis  preceded  by  the  minus  sign 
(the  subtrahend).     For  example: 

(3a-2b)-(2a+3b)=3a-2b-2a-3b  =  a-5b. 

62  RULE:    Parentheses  preceded  by  minus  signs  may  be  removed  if 

the  sign  of  each  term  enclosed  is  changed. 

Parentheses  preceded  by  plus  signs  may  be  removed  without  any 
change  of  sign. 

NOTE:  The  sign  preceding  a  parenthesis  disappears  when  the  paren- 
thesis is  removed. 

63  Other  signs  of  grouping  often  used,  are  the  brace  {  }  the 
bracket  [  ]  and  the  vinculum  .  These  have  the  same 
meaning  as  parentheses  and  are  used  to  avoid  confusion  when 
several  groups  are  needed  in  the  same  problem. 

64  When  several  signs  of  grouping  occur,  one  within  the  other, 
they  are  removed  one  at  a  time,  the  innermost  one  first  each 
time. 


50  REMOVAL   OF   PARENTHESIS 

Example :    Simplify  4x  —  {3x  +  ( —  2x  —  x  -  a)  } 

4x-|3x+(-2x-x-a)  | 

=  4x  —  |3x+  ( —  2x  —  x-fa)  }  (removing  the  vinculum) 

=  4x—  (Sx  —  2x  —  x+a  }  (removing  the  parenthesis) 

=  4x  —  3x + 2x + X  —  a  (removing  the  brace) 

=  4x  —  a  (combining  like  terms) 

NOTE:  In  the  case  of  the  vinculum,  special  care  must  be  taken. 
— X — a  is  the  same  as  —  (x— a).  The  minus  sign  preceding  the  vinculum 
is  not  the  sign  of  x. 

Exercise  38 
Simplify : 

1.  -6+{5-(7+3)  +  12} 

2.  10-[(7-4)-(9-7)] 

3.  4x-[2x-(x+y)+y] 

4.  -llb+[8b-(2b-fb)-3b] 


5.  8kz-[7kz-3kz-5kz] 

6.  x2-{'3x2-2F+l} 

7.  a'  -  ( -  6a2  -  12a +  8)  -  (a^^-  12a) 


8.  [6mn2- (8P-mn2+3n3  -  mn^) - (22mn2-8P)] 

9.  3a-(5a-{-7a+[9a-4]}) 

10.  c-[2c-(6a-b)-{c-5a+2b-(-5a+6a-3b)}] 
Solve  and  check: 

11.  4x-(5x-[3x-l])=5x-10 

12.  12x-{8--(8x-6)-(12-3x)}  =  0 

13.  -(12-20x)-{l92-(64x-36x)-12}  =  96 

14.  5x-[8x-{48-18x-(12-15x)}]  =  6 

15.  12x-[-81-(-27x-4+  10x)]  =  61- 

{8x-(-20-29-4x)} 


MULTIPLICATION 

Multiplication  of  Monomials 


zs^ 

Fig.  41 

65  Two  boys  of  equal  weight  are  on  a  teeter-board  at  equal 
distances  from  the  turning  point  (as  at  A  and  B,  Fig.  41). 
The  board  balances.  If  one  boy  weighed  one-half  as  much  as 
the  other,  he  would  have  to  be  twice  as  far  from  the  turning 
point  in  order  to  balance  the  other.  Similarly,  if  one  weighed 
one-third  as  much  as  the  other,  he  would  have  to  be  three 
times  as  far  from  the  turning  point  in  order  to  balance  the  other. 

From  these  illustrations,  it  is  readily  seen  that  a  weight  of 
one  pound,  four  feet  from  the  turning  point,  will  turn  the 
board  with  four  times  as  much  power  as  a  weight  of  one  pound, 
one  foot  from  the  turning  point.  A  weight  of  three  pounds, 
four  feet  from  the  turning  point,  will  turn  the  board  with  three 
times  as  much  power  as  a  weight  of  one  pound,  four  feet  from 
the  turning  point,  and  therefore  with  twelve  times  as  much 
power  as  a  weight  of  one  pound,  one  foot  from  the  turning 
point.  The  tendency  of  the  board  {lever)  to  turn  under  such 
conditions  is  called  the  leverage,  the  weights  acting  upon  it  are 
called  forces,  and  the  distance  of  the  forces  from  the  turning 
point  {fulcrum)  are  called  arms.  From  this  explanation  it  is 
evident  that: 

66  The  leverage  caused  hy  a  force  is  equal  to  the  force  times  the  arm. 

This  law  affords  a  very  convenient  means  of  working  out 
the  law  of  signs  for  multiplication  of  positive  and  negative 
numbers. 

51 


52  MULTIPLICATION 

67  Let  it  be  required  to  represent  the  product  of  (+5)  (+4). 
If  the  result  is  to  be  thought  of  as  a  leverage,  the  (+5)  will  be 
the  force,  and  the  (+4)  the  arm.  In  discussing  positive  and 
negative  numbers  in  Arts.  45,  46,  and  47,  measurements  up- 
ward and  to  the  right  were  represented  by  (+),  and  measure- 
ments downward  and  to  the  left  by  (  — ).  Then  the  (+5)  will 
be  considered  an  upward  pulling  force,  and  the  (+4)  an  arm 
measured  to  the  right  of  the  fulcrum  (Fig.  42) .     An  upward 


Fig.  42 


force  on  a  right  arm  causes  the  lever  to  turn  in  a  counter- 
clockwise (opposite  the  hands  of  a  clock)  direction.  To  be 
consistent  with  arithmetic,  (+5)  (+4)  must  be  (+20).  There- 
fore, in  determining  the  sign  of  the  result  of  multiplication,  a 
counter-clockwise  motion  of  the  lever  must  be  positive,  and  a 
clockwise  motion,  negative. 

68  Let  it  be  required  to  represent  the  product  of  (  —  5)  (—4). 
If  the  result  is  to  be  thought  of  as  a  leverage,  the  ( —  5)  will  be 
a  downward  pulling  force,  and  the  (—4),  a  left  arm  (Fig.  43). 


P^ 


Fig.  43 


It  is  seen  that  the  lever  turns  in  a  counter-clockwise  direction 
which  is  positive.     Therefore  (  —  5) (—4)  = +20. 


LAW   OF   SIGNS  53 

69  Let  it  be  required  to  represent  the  product  of  (+5)  (—4). 
In  this  case  there  is  an  upward-puWing  force  (+5),  on  a  left 


t-^ 


Fig..  44 


arm  (—4)  (Fig.  44).     The  lever  turns  in  a  clockwise  direction 
which  is  negative.    Therefore  (+5)(— 4)=  —  20. 

70  Let  it  be  required  to  represent  the  product  of  (  —  5)  (+4). 
In  this  case  there  is  a  downward-pulling  force  (  —  5),  on  a  right 


^ 


Fig.  45 


arm  (+4)  (Fig.  45).     The  lever  turns  in  a  clockwise  direction 
which  is  negative.    Therefore  (  —  5)  (+4)=  —20. 

From  the  four  preceding  articles: 

1.  (+5)(+4)  =  +20 

2.  (-5)(-4)  =  +20 

3.  (+5)(-4)=-20 

4.  (-5)(+4)  =  -20 

From  these  the  law  of  signs  for  multiplication  can  be  derived. 

71   Law  of  Signs  for  Multiplication:     If  two  factors  have  like 
signs,  their  product  is  plus. 

If  two  factors  have  unlike  signs,  their  product  is  minus. 


54  MULTIPLICATION 

Exercise  39 

Multiply: 

1.  (+|)(+i)  8.     (+3f)(+li) 

2.  (-f)(-i)  9.     (-6f)(-6j) 

3.  (+!)(-i)  10.     (7|)(-A) 

4.  (-f)(+i)  11.    (-i.i)(+i.i) 

6.     (-§)(+■!)  12.     (-2.03)(-4.2) 

6.  (+|)(-|)  13.     (+.3)(-.03) 

7.  (-|)(-t\)  14.     (+8.75)(+3i) 

15.     (-8.66)(-2i) 

72  By  Art.  21,  x^  means  x-x-x-x-x  and  x'  means  x-x-x. 

Therefore  (x^)(x^)  =  (x-x-x- x-x) (x-x-x)  =x^ 

From  this  the  law  of  exponents  for  multiplication  can  be 
derived. 

73  Law  of  Exponents  for  Multiplication:     To  multiply  powers 
of  the  same  base,  add  their  exponents. 

NOTE:    The  product  of  powers  of  different  bases  can  be  indicated 
only.     (x*)(y3)=x^y'. 

Example:    Multiply  (Ta^bx^)  by  (-3ab3y2) 
7a2bx3  =  7-a2.b-x3 
-3ab3y2=-3-a-b3-y2 
(7a2bx3)(-3abV)=7.a2-b-x3.(-3).a.b3.y2 
which  may  be  arranged  7-(  — 3)-a2-a-b-b^-x^-y2=  —  21a^b^xy. 

7^  RULE:  To  multiply  monomials,  multiply  the  numerical  coefficients, 
and  annex  all  the  different  bases,  giving  to  each  an  exponent 
equal  to  the  sum  of  the  exponents  of  that  base  in  the  two  factors. 


MULTIPLICATION 

OF   MONOMIALS 

Exercise  40 

Multiply: 

1. 

(3a3)(7a«) 

11. 

(+2ic)(-2id) 

2. 

(4x4)  (-6x5) 

12. 

(-9m2n3)(-7mV) 

3. 

(-5jmio)(-2m) 

13. 

(-7)(+m2n2) 

4. 

(-13a2b)(-2ab3) 

14. 

(+7|)(-gt2) 

5. 

(+5a2bc2)(-4abV) 

15. 

(3xy)(xy) 

6. 

(-6a2b)(+3b2c) 

16. 

(6r2)(-|r2) 

7. 

(+2ab)(-3cd) 

17. 

(_x2)(-x3) 

8. 

(+3)(-x) 

18. 

(-6.241)(+3.48m) 

9. 

(+5)(+|) 

19. 

(x  +  y)3.(x+y)4 

10. 

("DC-pq) 

20. 

(m2-n2)5.(m2-n2)2 

55 


Solve  and  check: 

21.  (x)(3)  =  (-3)(-6) 

22.  (r)(-2)  +  (r)(+3)  =  (-4)(-9) 

23.  (-3)(-6)  =  (w)(+2)  +  (0)(-5) 

24.  (-2)(d)  +  (d)(+3)  +  (2)(-3)=0 

25.  (+3)(-8)  +  (3k)(4)  +  (-2)(4k)  =  (0)(4) 

26.  (3s)(-6)-(-90)(-2)  =  (6s)(-6) 

27.  (3x)(+3)-(800)(+13)  =  (-5x)(19) 

28.  (5y)(-4)  +  (-56)(-l)  =  (-3y)(4) 

29.  (-9)(-2i)-(8x)(2)  =  (4i)(6)  +  (-7)(4x)  +  (6x)(0) 

30.  (-ll)(-3x)-(4|)(+6)  =  (20j)(-2)-(-3)(17x) 


56  MULTIPLICATION 

75  1.     What  is  the  leverage  caused  by  a  force  +7  on  an  arm 

-3? 

2.  What  is  the  leverage  caused  by  a  force  —16  on  an  arm 
+4? 

3.  What  is  the  leverage  caused  by  a  force  +3f  on  an 
arm  +6? 

4.  What  is  the  leverage  caused  by  a  force   —27  on  an 
arm  -2i? 

5.  What  is  the  leverage  caused  by  a  downward  force  of 
6,  on  a  right  arm  of  3? 

6.  What  is  the  leverage  caused  by  a  downward  force  of 
12,  on  a  left  arm  of  7? 

7.  What  is  the  leverage  caused  by  an  upward  force  of 
16,  on  a  left  arm  of  3 J? 

8.  What  is  the  leverage  caused  by  an  upward  force  of 
3  J,  on  a  right  arm  of  1^? 

76  Suppose  two  or  more  forces  are  acting  on  the  lever  at  the 
same  time  as  in  Figs.  46,  47,  48. 


-« — 2  — *-+-* 3 *- 

; i 

6  4 


Fig.  46 

What  is  the  leverage  caused  by  the  force  (  —  6),  Fig.  46? 

What  is  the  leverage  caused  by  the  force  (—4)? 

In  which  direction  will  the  lever  turn? 

This  may  be  expressed  by  (-6)(-2)H-(-4)(+3)  = 

+  12-12  =  0 


LAW   OF  LEVERAGES 

'^       ! 

J 

57 


Fig.  47 

What  is  the  leverage  caused  by  the  force  (  —  6),  Fig.  47? 
What  is  the  leverage  caused  by  the  force  (+4)? 
In  which  direction' will  the  lever  turn? 

This  may  be  expressed  by  (-6)(+2)  +  (+4)(+3)  = 
-12+12  =  0. 


^ 


Fig.  48 


What  is  the  leverage  caused  by  (  —  2),  Fig.  48? 
What  is  the  leverage  caused  by  (+3)? 
What  is  the  leverage  caused  by  (—3)? 
In  which  direction  will  the  lever  turn? 

This  may  be  expressed  by(-2)(-6)  +  (+3)(+l)  +  (-3)(+5) 
=  12+3-15  =  0. 

From  these  illustrations  the  law  of  leverages  may  be  derived. 

77  Law  of  Leverages:    For  balance,  the  sum  of  all  the  leverages 
must  equal  zero. 


58 


MULTIPLICATION 

Exercise  41 


1-10.     Find  the  unknown  force  or  arm  required  for  balance 
in  the  levers  shown  in  Figs.  49,  50,  51,  52,  53,  54,  55,  56,  57,  58. 

See  Art.  16. 


^ 


Fig.  49 


7     i 


^ 


Fig.  54 


r^-^  rr^^i — i 


Fig.  60 


Fig.  55 


F 


/^ 


r^i* 


Fig.  61 


A5" 


-J* 


r~^ — AT 


-« 10 


^0 


Fig.  56 


6 — -4— J 


-^         /^  JW 


Fig.  52 


260 

Fig.  67 


n 


^ 


r 


'i^-2i-\ 3i-*\ 


Fig.  53 


Fig.  68 


zyc 


11.  What  weight  12"  to  the  left  of  the  fulcrum  will  balance 
a  weight  of  10  lbs.,  9"  to  the  right  of  the  fulcrum?  (Draw  a 
figure). 


MULTIPLICATION   OF   SEVERAL   MONOMIALS  59 

12.  Two  boys  weighing  75  lbs.  and  105  lbs.  play  at  teeter. 
If  the  larger  boy  is  5'  from  the  fulcrum,  where  would  the 
smaller  boy  have  to  sit  to  balance  the  board? 

13.  A  crowbar  is  6'  long.  What  weight  could  be  raised  by 
a  man  weighing  165  lbs.,  if  the  fulcrum  is  placed  8"  from  the 
other  end  of  the  bar? 

14.  A  lever  12'  long  has  the  fulcrum  at  one  end.  How 
many  pounds  3'  from  the  fulcrum  can  be  lifted  by  a  force  of 
80  lbs.  at  the  other  end? 

15.  A  man  uses  an  8'  crowbar  to  lift  a  stone  weighing 
1600  lbs.  If  he  thrusts  the  bar  1'  under  the  stone,  with  what 
force  must  he  lift  to  raise  it? 


Multiplication  of  three  or  more  Monomials 

78  Example:    Multiply  (-2a)(-3a2)(+4a^)(-7a3) 
(-2a)(-3a2)  =  +6a3 
(+6a3)(+4a5)  =  +24a8 
(+24a8)(-7a3)=-168aii 
or  (-2a)(-3a2)(+4a5)(-7a3)=  -168a" 

Exercise  42 

Multiply: 

1.  (-3)(-4)(+5) 

2.  (-|)(+3i)(-i) 

3.  (+6)(-li)(-ft)(-7) 

4.  (Ila)(-7ab)(+4abc)(-9b2c2) 

5.  (aV)(-4a2b)(-lla3bO 


60 


MULTIPLICATION 


6.  (-4jab)(-3fac)(-3^bc)(Jabc) 

7.  (1 .25m2x)  ( -  2.4m3x2y)  ( -  4.63mxy2) 

8.  (-3.57)(+a^2)(_i|.aV). 

9.  4(x-y)3.  (x-y)2.{-7(x~7)} 

10.  3  (m2-n3)     {-4(m2-n3)4}  .   (m2-n'^)2 

Multiplication  of  Polynomials  by  Monomials 

a ..J-. — 1~ 


2D 


2b 


Fig.  59 


19  The  product  of  2(a+b)  may  be  represented  by  a  rectangle 
(Fig.  59)  having  a+b  for  one  dimension  and  2  for  the  other. 
The  area  of  the  entire  rectangle  is  equal  to  the  sum  of  the  two 
rectangles,  2a  and  2b,  or  2(a+b)  =  2a+2b. 

80  RULE:  To  multiply  a  polynomial  by  a  monomial,  multiply  each 
term  of  the  polynomial  by  the  monomial,  and  write  the  result  as 
a  polynomial. 

Example:     Multiply  Sm^  — 5mH-7  by  —  9m^ 
-9mH3m2-5m+7)=  -27m5-l-45m4-63m^ 


Exercise  43 

1. 

a3-7a2b+9ab2          by 

3a2b3 

2. 

6x5- 5x6 -7x*            by 

-7x» 

3. 

-3m2-n2+5mn       by 

4m2n3 

.4. 

a3-2a2x+4ax2--8x3  by 

-2ax 

6. 

ia2-iab+ib2          by 

-ib 

MULTIPLICATION   OP   POLYNOMIALS   BY   MONOMIALS  61 

6.  3x^-15x2+24  by         -^x^ 

7.  ^+1-1  by        12 

2^3     6 

8.  -L-1+-  by         -30 
10     5     6 

^       2m    3p      7  ,  ^^ 

9. — -  —  —  by         10 

5        2      15 

Simplify : 

10.  -2.4a3zH2ja2x-3|xz+.125z3) 

11.  5(3x+2y)+4(2x-3y) 

12.  3x(4a-2y)-5x(3y-5a) 

13.  -5(3a-2)-3(a-6)+9(2a-l) 

14.  x(3x-l)-2x(7-x)-5(x2+2x-l) 

,5.     16(^+1) -24(|+| 


Exercise  44 

Solve  and  check: 

1.  2(5m+l)=3(m+7)-5 

2.  3(5x+l)-4(2x+7)  =  3 

3.  3-(x-3)=7-2x 

4.  10(m-6)  =  3(m-2)-5 

5.  15y2+lly(2-5y)+4(10y2-9)  =  19 

6.  ^_^  =  3 


SUGGESTION:    2(x+5)-(x+l)=12    (clearing   of   fractions).    The 
line  of  the  fraction  has  the  same  meaning  as  a  parenthesis.     See  Art.  62. 


62 


MULTIPLICATION 


„     x+5    x-10     , 

7.     — =4 

3  4 


8. 


1 


x-1 


9.     6a-— -3(2a-l)=i 
a  a 

10.     5x-+2x+6_,,i 
5x 


Multiplication  of  Polynomials  by  Polynomials 
—Cf — M- d 


za 


ac 


2b 


be 


Fig.  60 

81  The  product  of  (a+b)(c+2)  may  be  represented  by  a 
rectangle  whose  dimensions  are  a+b  and  c+2  (Fig.  60).  The 
area  of  the  entire  rectangle  is  equal  to  the  sum  of  the  four 
rectangles,  ac,  be,  2a,  and  2b,  or  (a+b)(c+2)  =  acH-bc+2a+2b. 
It  will  be  seen  that  the  first  two  terms  are  obtained  by  multi- 
plying a+b  by  c,  and  the  last  two  terms  by  multiplying  a+b 
by  2.     It  is  convenient  to  arrange  the  work  thus: 

a+b 

c+2 

ac+bc 

+2a+2b 
ac+bc+2a+2b 


MULTIPLICATION   OF  POLYNOMIALS   BY   POLYNOMIALS 


63 


S2  RULE:     To  multiply  a  polynomial  by  a  polynomial,  multiply  one 
polynomial  by  each  term  of  the  other  emd  combine  like  terms. 

Example :    Multiply  x^— x+lbyx— 3 


X    +1 


x  -3 


x^- 

3? 

-     X2+    X 

-3x2+3x- 

-4x2+4x- 

-3 
-3 

Exercise  45 

Multiply: 

1.    x+2 

by 

x+7 

2.    a-3 

by 

a-5 

3.    m+2 

by 

m— 4 

4.     3m -5n 

by 

4m+3n 

5.    2a2+3b5 

by 

5a?+4:V> 

6.    a+1 

by 

a^-l 

7.    a-3 

by 

b+7 

8.    2x2-5x+7 

by 

3x-l 

9.    4m2-3ms- 

s^          by 

m2-3s2 

10.     x^-x^+x^-x+l     by         x+1 


Exercise  46 
Simplify : 

1.  (a  — b+c)(a  — b  — c) 

2.  (2n2+m2+3mn)(2n2-3mn+m2) 

3.  (ix-iy)(|x+iy) 

4.  (x+3)(x-4)(x+2) 

6.     (x2+xy+y2)(x2-xy+y2)(x2-y2) 


64 


MULTIPLICATION 


6.  (2a+3b)(6a-5b)  +  (a-4b)(3a-b) 

7.  5(x-4)(x+l)-3(x-3)(x+2)  +  (x+l)(x-5) 
Solve  and  check: 

8.  (y-5)(y+6)-(y+3)(y~4)  =  0 

9.  (m+3)(m+2)  =  (m+7)(m-5)+50 

10.    3  (2x-4)(x+7)-2  (3x-2)(x+5)  =  5-(3x-l) 
4(x2H-3x+7) 


11. 


12. 


13. 


14. 


16. 


=  2x 


2x4-7 

x+3    "^""^^ 

(3x4-2)  (2x+3)^ 
(2x-l)(x-f4) 
(x-2)(x-3)_(x4-3)(x-4)     (x-4)(x-5) 

3  4  ~  12 

1     x(2x+l)_(2x-3)(3x-f4)     2x+17 
4+2  6  ~      4 


Exercise  47 

Mu] 
1. 

itiply: 
2m2-m-l 

by 

3m2+m-2 

2. 

2p3-3p2q+7pq2+4q5 

by 

4p-3q 

3. 

a^+b^+ab^-f-a^b 

by 

a^b-ab^ 

4. 

5-3a+7a2 

by 

4+12a2 

6. 

-4mn4-3m2-lln2 

by 

2m2-5n2+7mr 

6. 

-5m2+9+2m3-4m 

by 

5m2-l+6m 

7. 

3ax2-4ax3-5ax5 

by 

l-x+2x2 

8. 

p3_6p2_^12p-8 

by 

p3+6p2+12p+8 

9. 

s3-2s2-s-l 

by 

s3+2s2-s+l. 

10. 

a-l+a^-a^ 

by 

1+a 

MULTIPLICATION   OF   POLYNOMIALS   BY   POLYNOMIALS        65 

11.  4x3-3x^+2x2-6  by  x-x^+l 

12.  3b3-7b2c+8bc2-c3  by  2b3H-8b2c-7bc2+3c3 

13.  a^+b^+c^+ab— bc+ac  by  a  — b  — c 

14.  a3-3+2a2-a  by  3-a+a3-2a2 

15.  ia-|b+fc-|d  by  ^a+fb-Jc+|d 

16.  |a2-fab-|b2  by  ija^-fb^ 

17.  2fm2n2-4jn3  by  Im^-fn 

18.  1.25a+2.375bH-3.5c  by  8a-8b+8c 

19.  .35a2+.25ab+3.75b2  by  4.1a2-.02ab-.57b2 

20.  3.5x2-2.1xy-1.05y2  by  4x-f 

Exercise  48 

Simplify : 

1.  (a-l)(a-2)(a-3)(a-4) 

2.  (a-b)(a2+ab+b2)(a3+b3) 

3.  (3x-4y)(2x+3y)(4x-5y)(x-7y) 

4.  (m+n)(m-n)(5^+^) 

6.  (x+y)(x3+y3){x2-y(x-y)} 

6.  (x+yH-z)(x-y+z)(x+y-z)(y+z-x) 

7.  (2a+5b-c-4d)2 

8.  (fa3-|b2)3 

2     3    4 

10.     (x+y)(x2-y2)-(x-y)(x2+y2) 


66  MULTIPLICATION 

11.  (3a-2b)(2a2-3ab+2b2)-3a(2a2-3ab) 

12.  6(m-n)(m+n)-4(m2+n2) 

13.  12(x-y)-(x2+x-6)(x2+x+y) 

14.  15ab-3(2a2+4b2)  +  (3a-2b)(5a-3b) 

15.  6(a+2b-2c)2-(2a+2b-c)2 

16.  (x2+l)(x-l)-(x-3)(2x-5)(x+7)-(x+2)3 

17.  (a+b+c)3-3(a+b+c)(a2+b2+c2) 

18.  (2in2-3mn+4n2)K^-^)^ 

a+b+c  a-b+c  a+b-c  b+c-a 
2*2*2  2~ 

(x-2)(2x-3)    (x+2)(2x+3)    (x^+ 4)  (4x^^+9) 
3*7*2 


19. 


20. 


Exercise  49.     (Review) 
Solve  and  check: 
1.     (-16)(-x)  +  (-13)(+12)  +  (-2)(+2x)=0 


|)  +  (-14)(-|)  +  (-10f)(+i|; 


2.     (+15)(-^)  +  (-14)(-^)  +  (-10f)(+±|)  =  0 


3.  (-4f)(5x)  +  (+7|)(7x)  +  (-8f)(0)  +  (7)(-12)=0 

4.  3x-3(ix-7)=35 

5.  (2x-l)(3x+7)-3x2=(x-l)(3x-12)+20 

^  3x+5     ,     x-7 

^-  "^— ^"~6~ 

7.  (4x+f)(fx-i)=|i 

^  3(3 -2x)     2(x-3)         2     4(x+4)    .    1 

^'  10       ~       5        '^'^^-~^       +10 

9.  t(x+5)-|-(x+7)+V(x+l)-|(2x-5)=i(x+22) 


EQUATIONS   INVOLVING   MULTIPLICATION  67 

(x-l)(x+2)_(2x+l)(x+2)     (2x+l)(x-l) 
2  12  ~  6 

11.  If  I"  the  supplement  of  an  angle  is  subtracted  from  the 
angle,  the  result  is  27°.     Find  the  angle. 

12.  If  f  the  complement  of  an  angle  is  subtracted  from 
three  times  the  angle,  the  result  is  39°.     Find  the  angle. 

13.  If  -f  of  the  supplement  of  an  angle  is  decreased  by  f 
of  the  complement,  the  result  is  53°.     Find  the  angle. 

14.  J  the  supplement  of  an  angle  is  equal  to  the  angle 
diminished  by  f  of  its  complement.     Find  the  angle. 

15.  Find  three  consecutive  numbers  such  that  the  product 
of  the  second  and  third  exceeds  the  product  of  the  first  and 
second  by  40. 

16.  The  difference  of  the  squares  of  two  consecutive  num- 
bers is  43.     Find  the  numbers. 

17.  The  length  of  a  rectangle  is  three  times  its  width.  If 
its  length  is  diminished  by  6,  and  its  width  increased  by  3, 
the  area  of  the  rectangle  is  unchanged.     Find  the  dimensions. 

18.  Two  weights,  123  and  41  respectively,  are  placed  at 
the  ends  of  a  bar  24  ft.  long.  Where  should  the  fulcrum  be 
placed  for  balance?  (Suggestion:  Let  x  =  one  arm,  24— x  = 
the  other.) 

19.  A  man  weighing  180  lbs.  stands  on  one  end  of  a  steel 
rail  30  ft.  long,  and  finds  that  it  balances  with  a  fulcrum  placed 
2  ft.  from  the  center.  What  is  the  weight  of  the  rail?  (Sug- 
gestion: The  weight  of  the  rail  may  be  considered  a  down- 
ward force  at  the  middle  point  of  the  rail.) 

20.  An  I-beam  32  ft.  long  weighing  60  lbs.  per  foot,  is 
being  moved  by  placing  it  upon  an  axle.  How  far  from  one 
end  shall  the  axle  be  placed,  if  a  force  of  213^  lbs.  at  the  other 
end  will  balance  it? 


DIVISION 


Division  of  Monomials 

83  To  divide  positive  and  negative  numbers,  a  law  of  signs  and 
a  law  of  exponents  are  necessary.  Tiiese  may  be  derived  from 
the  same  laws  for  multiplication,  from  the  fact  that  the  product 
divided  by  one  factor  equals  the  other  factor. 

By  Art.  70:        1.  (+5)(+4)  =  +20 

2.  (-5)(-4)  =  +20 

3.  (+5)(-4)  =  -20 

4.  (-5)(+4)  =  -20 
(+5)  =  +4 
(+4)  =  ? 
(-5)= -4 
(-4)  =  ? 
(+5)  =-4 
(-4)  =  ? 
(-5)  =  +4 
(+4)  =  ? 


Therefore,  from  1. 


from  2 


l(+20)- 
l(+20)- 
/(+20)- 
l(+20)- 

(-20)- 

(-20)- 

(-20)- 

(-20)- 

84  Law  of  Signs  for  Division: 
their  quotient  is  plus. 

If  two  numbers  have  unlike  signs,  their  quotient  is  minus. 


from  3 


from  4 


//  two  numbers  have  like  signs. 


Divide : 

1.     (+f)-(+i) 

2.  (-i)-(+i) 

3.  (+i)^(-i) 

4.     (-l)-(-l) 


(-f+)-(+3V) 


Exercise  50 

6. 
7. 
8. 
9. 
10. 
68 


(+2i) 
( 


16\ 
2  l) 


(+3f) 
■( 


(+7i)-( 


10) 

9) 

(-42)  +  (-fi) 
(+72)  ^(-41) 


DIVISION    OF   SIGNED    NUMBERS  69 

11.  (-8.5)-^(-1.7)  16.  (+3.6)^(-2i) 

12.  (+3.2)-(-.8)  17.  (-3i-56)^(-6.25) 

13.  (+2.65)  ^(+100)  18.  (-34.56) -^  (-.288) 

14.  (-.008)-^(-.02)  19.  (+26i)-^(-llf) 

15.  (-15)^(+.003)  20.  (-.0231)-^(-6|) 

Exercise  51 
Solve  and  check: 

1.  3x+14-5x+15  =  4x+ll 

2.  20x+15+32x+193-12  =  36x+100-32x 
•     3.     s(2s-3)-2s(s-7)+231  =  0 

4.  (x-5)(x-6)  =  (x-2)(x-3) 

5.  (3x-l)(4x-7)  =  12(x-l)2 
e      12     3x       i9_4x     172 

x+3_x-2     3x-5     1 
2    ~    3    ~    12    "^"4 

2x±5    x-3     x_  1 

®-         9     ~    5    -3-^^+173 

9.     f(x+2)-T'5(x+5)  +  10  =  2-i(x+l) 

s(s-2)_s(s-9)     -2s^-91 
"•5  3      ~        15 

85  By  Art.  72,  {x^){x^)  =  x^ 

Therefore      (x^)  4- (x^)  =  x^ 
(x8)^(x3)  =  ? 

86  Law  of  Exponents  for  Division:   To  divide  powers  of  the  same 
base,  subtract  the  exponent  of  the  divisor  from  that  of  the  dividend. 


/  U  DIVISION 

NOTE:    The  quotient  of  powers  of  different  bases  can  be  indicated 
only. 

(x^)-(y3)=^ 

Example:    Divide  48  a^b^c^x^V  by  -Sa^b^c^x^ 

—  •;  =  -6.a5.b.l.x3.y4 

—  Sa^b^c^x^  "^ 

=  —  6a%xy 

c*  c" 

NOTE :     —  =  1  •     Also—  =  c"  by  the  law  of  exponents. 

Therefore  c°  =  1  and  may  be  omitted  as  a  factor  in  problems  like  the 
above  example. 

S7  RULE:  To  divide  a  monomial  by  a  monomial,  divide  the  numerical 
coefficients,  and  annex  all  the  different  bases,  giving  to  each  an 
exponent  equal  to  the  difference  of  the  exponents  of  that  base 
in  the  two  monomials. 


Exercise  52 


Divide: 

1.  -91a 

2.  -32x6 

3.  +22a^b3c7 

4.  +6jm3n3 

6.  -8jpi3qi'r'^ 

6.  -4.24xV^z7 

7.  1.75an)«x5 

8.  -.SSm^n^ 

9.  -.OOlxVm^ 
10.  +3.1416a2b3mi« 


by  +13 

by  -8x^ 

by  -lla^b^c^ 

by  —  l^mn^ 

by  Sfqior^ 

by  —  .4xy  z^ 

by  -.35x^ 

by  17n* 

by  —  lOOxym 

by  +4a2b3mi« 


DIVISION   OF   MONOMIALS  71 

Simplify : 

1155a^x7z5  3.1416r^ 

-231a2x«z  .7854r2 

-1.732t^uV  +a^bVQ 

+2u6s2  -2a3bc» 

3.1416xyV7  -a^(x+y)« 

".        _iix26y225  "•     +2-fa(x+y)4 

27m^n^x7  (a+b)^(a+b)^(x+y)^ 

1.125m2n3x7  .0625(aH-b)3(x+y)^ 

32.16t^  18.75a(m^-n^)^Q 

-.08t2  ^"*        2i(m2-n^)7 


Exercise  53 
Solve  for  x  and  check: 

1.  -ax=-ab  6.    3a2b3x=-12a3b3 

2.  +|bx=-8b  7.    7a-3ax  =  28a 

3.  —  .3inx  =  2.4m  8.    4mx— 7mx=12m  — 18m 

4.  -4x=-12(a+b)  9.    6a2b-7ax= -29a2b 

.n      X     3      1 

5.  -fx  =  2m  10.     3^--  =  e^ 

lOx      .  5x     5m 

12.  ^b_3(x-2b)^^^ 

o  o 

13.  (x-5y)(x+4y)=x2+y2 

14.  (x+m)(x-2m)-x(x-7m)=m(3x-5m) 
(x-3s)(x-2s)     x(x-5s)     x(x-3s) 

15.      7Z 71 = :; • 


72  DIVISION 

Division  of  a  Polynomial  by  a  Monomial 

88  By  Art.  79,  2(a+b)  =  2a+2b 

Therefore, — '- =  a+b 

89  RULE:     To  divide  a  polynomial  by  a  monomial,  divide  each  term  of 

the  polynomial  by  the  monomial,  and  write  the  result  as  a  poly- 
nomial. 

Example:     Divide  2 lm«-35m^+7m3        by  -7m2 
21m«-35m4+7m2 


-7m2 


=  -3m*+5m2-m 


Exercise  54 
Divide: 

a^x'c* — ax^c^y^ + a^xc^z 

axc^ 
4xVV-  12xVz^-24xVz^H-16xyz 

— 4xyz 
2.31m2n2+7.7m3n3-  .33m%^ 


l.lm^n^ 

-  ijt  V  -  9.81tV  - .  378tv^ 

-9tv2 
1  ■  125a^x^z^  -  .375aVz^  -  4.2a^x^z^ 
.25aVz2 

-3fabcd+2|bcdeH-7|acde 
-l|cd 

Solve  for  x  and  check: 

7.  ax  =  2ab  — 3ac+4ae 

8.  3a2m3x  =  I.lla3m3-3.3a2m* 

9.  4m2s2x  -  3.2m3s2  =  ISam^s^ 


6. 


EQUATIONS   INVOLVING   DIVISION  73 

10.  3jxyz-1.4y2z  =  .35yz2-70yz 

11.  4m2x-7m3n4-3m2x+8m2n5  =  5m3n4-2m2x+2m2n5 

12.  il^-T^B^^^.lOlm^ 

o  u 

^^      X     a2b+2bx 

13.  -  — ^ =  3ab2 

a  ab 

nx     3(n2x-m2n2)      4n 

14.  8mn + -^ = hm^n 

m  mn  m 

2mx+a2m^     5(b^n^+nx)_^       2m2n-3mn^ 

15.      — — <jX 

m  n  mn 


Division  of  Polynomials  by  Polynomials 

90  By  Art.  81,  (a+b)(c+2)  =ac+bc+2a+2b 

a+b 

In  multiplying  a+b  by  c  + 2,  the  first  two  terms  were  obtained 
by  multiplying  a+b  by  c,  and  the  last  two  by  multiplying 
a+b  by  2.     In  dividing,  the  c  may  be  obtained  by  dividing  ac 
by  a,  and  the  2  may  be  obtained  by  dividing  2a  by  a.     It  is 
convenient  to  arrange  the  work  as  follows: 
c  +  2 
a+b)ac+bc+2a+2b 
ac+bc 

+2a+2b 
+  2a+2b 

9t  RULE:  To  divide  a  polynomial  by  a  polynomial,  divide  the  first 
term  in  the  dividend  by  the  first  term  in  the  divisor  to  obtain  the 
first  term  of  the  quotient.  Multiply  the  divisor  by  the  first  term 
of  the  quotient,  and  subtract  the  result  from  the  dividend.  To 
obtain  the  other  terms  of  the  quotient,  treat  each  remainder  as 
a  new  dividend  and  proceed  in  the  same  way. 


74  DIVISION 

Example  (1):    Divide  a3-6a2-19a+84  by  a-7. 
a^+a-12 
a-7)a3-6a2-19a+84 
a^-7a^ 
+a2-19a 
+a-—  7a 


-12a+84 
-12a+84 

Example  (2):     Divide  24+26x3+120x4- 14x- 11  Ix^ 

by  -x-6+12x2 

NOTE:     Arrange  the  terms  according  to  the  powers  of  x,  in  both 
dividend  and  divisor. 

10x2+3x-4 


12x2-x-6)120x4+26x3- 

-  111x2- 14x -1-24 

120x^-10x3- 

-60x2 

+36x3- 

-  51x2- 14x 

+36x3- 

-     3x2 -18x 

- 

-  48x2+  4x+24 

- 

-  48x2+  4x+24 

Exam 

ple  (3):    Divide 

i  x^+xV- 

f  y4  by  x2+xy+y' 

x2- 

xy +y2 

x2+xy+y2)x* 

+xV 

+y^ 

x4+ 

x3y+xV 

- 

x3y 

+y^ 

- 

X3y_x2y2 

—  xy3 

+xy 

+xy3+y4 

+xV 

+xy3+y4 

Exercise  55 

Divide  the  following: 

1. 

x2-7x+12 

by    x-3 

2. 

a2-2a-15 

by    a  — 5 

3. 

a2-3ab-28b2 

by    a+4b 

DIVISION   OF   POLYNOMIALS   BY   POLYNOMIALS 


75 


4. 

6m4-29m2+35 

by    2m2-5 

6. 

12x2+31xy-15y2 

by    x+3y 

6. 

m3+3m2-13m-15 

W  m+1 

7. 

10x3-  19x2y+26xy2-8y3 

by  2x2-3xy+4y2 

8. 

3m4  -  lOm^  -  16m2  -  10m  -  3 

by  3m2+2m+l 

9. 

2x^-x3y+4xy+xy3+12y4 

by  x2-2xy+3y2 

10. 

4x4-24x3+51x2-46x+15 

by  2x2-7x+5 

11. 

9xV  -  15x3y2+  13xy  -  3xy4 

by  3x2y-xy2 

12. 

Sm^n  -  22m6n2  -  7m^n3+ 

53m4n4-30m3n5 

by  4Tn4+3m3n-5m2n2 

13. 

10x3  _  29.3x2y +37xy2  -  20y3 

by  2.5x2 -4.2xy+4y2 

14. 

14x3+17x2y+39xy2+17y3 

by  3.5x2+2.5xy+8.5y2 

15. 

18x3-53x2+27x+14 

by  4x-8 

16. 

13.12m5n+1.36m%2_ 

7.15m3n3+2.35m2n4-.125mn 

'  by  3.2in2n-2.4mn2+.5n3 

17. 

a2+ab+2ac+bc+c2 

by  a+c 

18. 

a^bx + abcx + a^cx + ab  V + 

b^cy+abcy 

by  ax+by 

19. 

x2+8-10x+x3 

by  2+x2-3x 

20. 

m^ + n^ — 4m3n — 4mn3 + Gm^n^ 

'  by  m2+n2  — 2mn 

21. 

a5-9a3+7a2-19a+10 

by  a2+3a-2 

22. 

16m4-72m2n2+81n4 

by  4m2-12mn+9n2 

23. 

m^— 64n3 

by  m— 4n 

24. 

32as+243b5 

by  2a+3b 

25. 

a2+2ab+b2-c2 

by  a+b— c 

26. 

a2-_x2-2xy-y2 

by  a— x-y 

27. 

a^+4+3a2 

by  a2+2-a 

76  DIVISION 

28.  4a2-b2-6b-9  by  2a+b+3 

29.  a2-b2+x2-y2+2ax+2by         by  a+x+b-y 

30.  m2-2mn+n2+3m-3n+2       by  m-n+1 

Solve  and  check: 

31.  (a+3)x  =  ab-fa+3b+3 

32.  (a2-4ab+3b2)x  =  a3-8a2b+19ab2-12b' 

33.  (2m-3n)x  =  8m3-22m2n+mn2+21n3 

34.  (a+bH-2)x  =  a2+2ab+b2+4a+4b+4 

35.  (y-f-2)x  =  y3-y2-34y-56 


Exercise  56 

1.  One  number  is  6  more  than  another,  and  the  difference 
of  their  squares  is  144.     Find  the  numbers. 

2.  One  number  is  3  less  than  another,  and  the  difference 
of  their  squares  is  33.     Find  the  numbers. 

3.  Divide  42  into  two  parts  such  that  J  of  one  is  equal  to 
^  of  the  other. 

4.  Divide  57  into  two  parts  such  that  the  sum  of  ^  of 
the  larger  and  \  of  the  smaller  is  12. 

5.  The  difference  of  two  numbers  is  11,  and,  if  18  is  sub- 
tracted from  f  of  the  larger,  the  result  is  yof  the  smaller 
number.     Find  the  numbers. 

6.  Divide  24  into  two  parts  such  that  if  ^  of  the  smaller 
is  subtracted  from  f  of  the  larger,  the  result  is  9. 

7.  If  the  product  of  the  first  two  of  three  consecutive 
numbers  is  subtracted  from  the  product  of  the  last  two,  the 
result  is  18.     Find  the  numbers. 


REVIEW   PROBLEMS  77 

8.  If  the  square  of  the  first  of  three  consecutive  numbers 
is  subtracted  from  the  product  of  the  last  two,  the  result  is 
41.     Find  the  numbers. 

9.  I  paid  a  certain  sum  of  money  for  a  lot  and  built  a 
house  for  3  times  that  amount.  If  the  lot  had  cost  $240  less 
and  the  house  $280  more,  the  lot  would  have  cost  i  as  much 
as  the  house.     What  was  the  cost  of  each? 

10.  A  boy  has  2 J  times  as  much  money  as  his  brother. 
After  giving  his  brother  $25.00,  he  has  only  1^  times  as  much. 
How  much  had  each  at  first? 

11.  The  sum  of  J  a  certain  angle,  J  of  its  complement  and 
Y^Q-  of  its  supplement  is  48°.     Find  the  angle. 

12.  Three  times  an  angle,  minus  4  times  its  complement,  is 
equal  to  -j^y  of  its  supplement  +  131°.     Find  the  angle. 

13.  If  3  times  an  angle  is  subtracted  from  J  its  supplement, 
the  result  is  yt  of  its  complement. 

14.  A  certain  rectangle  contains  15  sq.  in.  more  than  a 
square.  Its  length  is  7  in.  more  and  its  width  3  in.  less  than 
the  side  of  the  square.     Find  the  dimensions  of  the  rectangle. 

15.  The  altitude  of  a  triangle  is  4  in.  more  than  the  base, 
and  its  area  exceeds  one  half  the  square  of  the  base  by  16. 
Find  the  base  and  altitude.  (Suggestion:  See  Exercise  15, 
problem  5.) 

16.  A  wheelbarrow  is  loaded  with  a  barrel  of  flour  weighing 
196  lbs.  The  center  of  the  load  is  2'  from  the  axle  of  the 
wheel.  What  force  at  the  handles,  4j'  from  the  axle  of  the 
wheel,  will  be  required  to  raise  the  load? 

17.  A  wheelbarrow  is  loaded  with  5  bars  of  pig  iron  weigh- 
ing 77  lbs.  each.  How  far  from  the  axle  of  the  wheel  should 
the  center  of  the  load  be  placed,  if  a  force  of  154  lbs.  4  ft.  from 
the  axle  will  raise  it? 


78  DIVISION 

18.  A  timber  12"  X 18"  X  24'  is  balanced  on  wheels  and  an 
axle  by  a  force  of  120  lbs.  at  one  end.  How  far  from  the  center 
shall  the  axle  be  placed  if  the  timber  weighs  45  lbs.  per  cu.  ft.? 

19.  A  lever  12'  long  weighs  24  lbs.  If  a  weight  of  30  lbs. 
is  hung  at  one  end  and  the  fulcrum  is  placed  4'  from  this  end, 
what  force  is  needed  at  the  other  end  for  balance? 

20.  A  piece  of  steel  1'  long,  weighing  15  lbs.  per  foot,  is 
resting  upon  one  end.  A  weight  of  1400  lbs.  is  placed  l|' 
from  that  end.  What  force  at  the  other  end  is  necessary  to 
balance  the  load? 


CHAPTER  V 
RATIO,  PROPORTION,  AND  VARIATION 

Ratio 

92  Ratio:  The  relation  of  one  quantity  to  another  of  the  same 
kind  is  called  a  ratio.  It  is  found  by  dividing  the  first  by  the 
second.  For  example:  the  ratio  of  $2  to  $3  is  f ,  written  also 
2:3;  the  ratio  of  7"  to  4"  is  |;  the  ratio  of  18"  to  6'  is  +|  =  i. 

93  Terms  of  Ratio:  The  numerator  and  denominator  of  a  ratio 
are  respectively  the  first  and  second  terms  of  a  ratio.  The 
first  term  of  a  ratio  is  called  its  antecedent,  and  the  second,  its 
consequent. 

Exercise  57 

1.  Find  the  ratio  of  85  to  51. 

2.  Find  the  ratio  of  27  to  243. 

3.  Find  the  ratio  of  2j  to  3f . 

4.  Find  the  ratio  of  6.25  to  87.5. 

5.  Find  the  ratio  of  y\  to  .3125. 

6.  Find  the  ratio  of  8"  to  6'. 

7.  Find  the  ratio  of  12a  to  16a. 

8.  Find  the  ratio  of  577-  to  Stt. 

9.  Find  the  ratio  of  a  right  angle  to  a  straight  angle. 

10.  Find  the  ratio  of  a  right  angle  to  a  perigon. 

11.  Find  the  ratio  of  a  straight  angle  to  a  perigon. 

12.  Find  the  ratio  of  f  of  a  perigon  to  -|  of  a  right  angle. 

79 


80  RATIO,    PROPORTION,   AND   VARIATION 

13.  Find  the  ratio  of  55°  to  its  complement. 

14.  Find  the  ratio  of  55°  to  its  supplement. 

15.  Find  the  ratio  of  45°  to  §  its  supplement. 

16.  Find  the  ratio  of  the  supplement  of  48°  to  its  comple- 
ment. 

17.  A  door  measures  4'  X  8'.  What  is  the  ratio  of  the 
length  to  the  width? 

18.  There  were  25  fair  days  in  November,  while  the  rest 
were  stormy.  What  was  the  ratio  of  the  fair  to  the  stormy 
days? 

19.  The  dimensions  of  two  rectangles  are  5"  X  8",  and  6" 
X  S''.  Find  the  ratio  of  their  lengths,  widths,  perimeters,  and 
areas. 

20.  The  bases  of  two  triangles  are  3.9  and  2.4,  and  their 
altitudes  are  respectively  .8  and  .7.  Find  the  ratio  of  their 
areas. 

21.  Find  the  ratio  of  the  circumferences  of  two  circles  whose 
diameters  are  respectively  5j"  and  2f ".  (See  Exercise  16, 
problem  2.) 

22.  Find  the  ratio  of  the  areas  of  two  circles  whose  diame- 
ters are  respectively  IV  and  13".     (See  Exercise  16.) 

23.  Find  the  ratio  of  the  two  values  of  P  in  the  formula  P  = 
awh,  when  a  =  120,  w  =  .32,  h  =  9j,  and  when  a  =  48,  w  =  .38, 
and  h  =  24. 

24.  Find  the  ratio  of  the  two  values  of  F  in  F=l§d+J, 
when  d  =  if,  and  when  d  =  2j. 

25.  Find  the  ratio  of  the  two  values  of  S  in  S  =  Jgt2,  when 
t  =  3j,  and  when  t  =  10j.     (See  Exercise  17.) 


RATIO    AS    DECIMALS 


81 


94  To  Express  Ratios  as  Decimals:  It  is  often  convenient  to 
have  results  in  decimal  rather  than  in  fractional  form.  For 
example :  the  ratio  -I-  is  often  written  .875. 


Exercise 

58 

Find  the  decimal  equivalents  of  the  following 

ratios: 

1.   A 

8 

3.      1 
25 

5.  ?-^ 

32 

7.     '1 
64 

9. 

.72 

1* 

2.      ^ 
16 

4.      1? 
20 

6.  !i 

30 

8      2f 

10. 

3.24 
129.e 

95  Sometimes  it  is  sufficiently  accurate  to  express  the  decimal 
to  two  places  only.  In  this  case  it  is  necessary  to  determine  the 
third  place,  and,  if  this  is  5  or  more,  it  is  customary  to  increase 
the  second  place  by  1.  For  example:  the  ratio  y|^=.946  +, 
which  would  be  written  .95  if  two  places  only  are  desired. 


Exercise  59 


Find  the  decimal  equivalents  of  the  following  ratios,  correct 
to  .01 : 


1. 


9 


10  ^19          ^25          ,     37.5 
—  3.      —  4.     —  6.     

11  16  7|  5.15 


Percentage  is  found  by  reducing  a  ratio  to  a  decimal  correct  to 
.01,  and  multiplying  it  by  100. 


For  example:     ^?^  =  6.688  =  669%. 
.0369 


82  RATIO,    PROPORTION,   AND   VARIATION 

6.  In  a  class  of  27  students,  22  passed  an  examination. 
Find  the  percentage  of  successful  students. 

7.  A  base  ball  player  made  89  hits  out  of  321  times  at  bat. 
Find  his  batting  average  (percentage). 

8.  The  total  cost  of  manufacturing  an  article  is  $5.36  of 
which  $2.79  represents  labor.  What  per  cent  of  the  total  cost 
is  the  labor? 

9.  If  62j  tons  of  iron  are  obtained  from  835  tons  of  ore, 
what  per  cent  of  the  ore  is  iron? 

10.  In  a  class  of  students,  25  passed,  2  were  conditioned,  and 
6  failed.     Find  the  percentage  of  failures. 

11.  Babbitt  metal  is  by  weight  92  parts  tin,  8  parts  copper, 
and  4  parts  antimony.     Find  the  percentage  of  copper. 

12.  Potassium  nitrate  is  composed  of  39  parts  of  potassium, 
14  parts  of  nitrogen,  and  48  parts  of  oxygen.  Find  the  per- 
centage of  potassium. 

13.  Potassium  chloride  is  composed  of  39  parts  of  potassium 
and  35.5  parts  of  chlorine.     Find  the  percentage  of  chlorine. 

14.  Baking  powder  is  composed  of  3j  parts  of  soda,  if  parts 
of  cream  of  tartar,  and  6.5  parts  of  starch.  Find  the  percentage 
of  cream  of  tartar. 

16.  If  12  quarts  of  water  are  added  to  25  gallons  of  alcohol, 
what  per  cent  of  the  mixture  is  alcohol? 

16.  If  5  lbs.  of  a  substance  loses  5  oz.  in  drying,  what  per 
cent  of  its  original  weight  was  water? 

17.  If  5  lbs.  of  a  dried  substance  has  lost  5  oz.  in  drying, 
what  per  cent  of  its  original  weight  was  water? 

18.  If  a  dried  substance  absorbs  5  oz.  of  water  and  then 
weighs  5  lbs.,  what  per  cent  of  its  original  weight  is  water? 


SPECIFIC   GRAVITY  ^  83 

19.  The  itemized  cost  of  a  house  is  as  follows: 

Masonry      .      .  $  750  Plumbing  .  .  .  $350 

Carpenter  Work  $  900  Furnace  .  .  .  $150 

Lumber        .      .  $1200  Painting  .  .  •  .  .  $300 

Plastering    .      .  $  250 

What  per  cent  of  the  total  cost  is  represented  by  each 
item? 

Check  by  adding  the  per  cents. 

20.  The  population  of  Detroit  in  1900  was  285,704,  and  in 
1910,  it  was  465,776.     Find  the  percentage  of  increase. 

96  Specific  Gravity:    The  specific  gravity  of  a  substance  is  the 

ratio  of  the  weight  of  a  certain  volume  of  the  substance  to  the 

weight  of  the  same  volume  of  water.     For  example:  if  a  cubic 

inch  of  copper  weighs  .321  lbs.,  and  a  cubic  inch  of  water  weighs 

321 
.0361  lbs.,  the  specific  gravity  of  copper  is =  8.88. 

Example.     The  dimensions  of  a  block  of  cast  iron  are  3j"X 
2f  "X 1",  and  its  weight  is  37.5  oz.     Find  its  specific  gravity. 

3jX2f  X  1  =8.94  cu.  in.   (the  volume  of  the  block) 

.0361  lbs.  =  .5776  oz.  (weight  of  1  cu.  in.  of  water) 
.5776  X  8 .  94  =  5 .  16   (weight  of  8.94  cu.  in.  of  water) 

— '—  =  7 .  27,  (specific  gravity  of  iron) 
5.16 

NOTE:    Specific  gravity  is  usually  found  correct  to  .01. 


Exercise  60 

1.     A  cubic  inch  of  aluminum  weighs  .0924  lbs.     Find  its 
specific  gravity. 


84  KATIO,    PROPORTION,    AND   VARIATION 

2.  A  cubic  inch  of  tungsten  weighs  .69  lbs.     Find  its  specific 
gravity. 

3.  A  cubic  inch  of  cast  steel  weighs  .282  lbs.     Find  its 
specific  gravity. 

4.  A  cubic  inch  of  lead  weighs  6.56  oz.     Find  its  specific 
gravity. 

5.  A  cubic  foot  of  bronze  weighs  550  lbs.     Find  its  specific 
gravity. 

6.  A  cubic  foot  of  cork  weighs  240  oz.     Find  fts  specific 
gravity. 

7.  A  brick  2"  X  4"  X  8"  weighs  4.64  lbs.     Find  its  specific 
gravity. 

8.  A  cedar  block  5"  X  3"  X  2"  weighs  10.5  oz.     Find  its 
specific  gravity. 

9.  Each  edge  of  a  cubical  block  is  2' .     If  it  weighs  4450 
lbs.,  what  is  its  specific  gravity? 

10.  A  man  weighing  185  lbs.,  displaces  when  swimming 
under  water,  5760  cu.  in.  of  water.  Find  the  specific  gravity  of 
the  human  body. 

97  Separating  in  a  given  ratio. 

Example:  Divide  17  into  two  parts  which  shall  be  in  the  ratio  f . 
Let  2x  =  one  part. 

3x  =  other  part.  mott?  ^'^-^ 

Then2x+3x  =  17  ^^       ii^S 

5x  =  17 
x  =  3f 
2x  =  6|-,  one  part. 
3x=10|-,  other  part. 

Check:    6|+10i=17,         T^  =  ^  =  f 


SEPARATING   IN   A   GIVEN   RATIO  85 

Exercise  61 

1.  Divide  20  in  the  ratio  f . 

2.  Divide  18  in  the  ratio  ^. 

3.  Divide  100  in  the  ratio  -f-. 

4.  Divide  200  in  the  ratio  ^. 

6.     Two  supplementary  angles  are  in  the  ratio  ^.     Find 
them. 

6.  Two  complementary  angles  are  in  the  ratio  -J.     Find 
them. 

7.  A  board  18"  long  is  to  be  divided  in  the  ratio  ^.     How 
far  from  each  end  is  the  point  of  division? 

8.  If  a  line  4'  6"  long  is  divided  in  the  ratio  ^,  what  is  the 
length  of  each  part? 

9.  Divide  a  legacy  of  $25,000  between  two  persons  so  that 
their  shares  shall  be  in  the  ratio  ^. 

10.  The  sides  of  a  rectangle  are  in  the  ratio  -J,  and  its 
perimeter  is  100.     Find  the  dimensions  of  the  rectangle. 

11.  Bronze  is  composed  of  11  parts  tin  and  39  parts  copper. 
Find  the  number  of  pounds  of  tin  and  copper  in  625  lbs.  of 
bronze. 

12.  A  gold  medal  is  18  carats  fine  (18  parts  of  pure  gold  in  24 
parts  of  the  whole  alloy).  Find  the  amount  of  pure  gold  in  the 
medal  if  it  weighs  2.7  oz. 

13.  Two  men  purchase  some  property  together,  one  paying 
$750  and  the  other  $450.  If  the  property  is  sold  for  $2,000, 
what  will  be  the  share  of  each? 

14.  Two  men  agree  to  do  a  piece  of  work  for  $45.  The  work 
is  completed  in  10  days,  but  one  of  them  was  absent  2  days. 
How  should  the  pay  be  divided? 


86  RATIO,    PROPORTION,    AND   VARIATION 

16.    How  much  copper  would  there  be  in  208  lbs.  of  Babbitt 
metal?     (See  Exercise  59,  problem  11.) 

16.  Divide  a  perigon  into  three  angles  in  the  ratio  7:8:9. 

17.  Divide  a  line  5'  3"  long  into  four  parts  in  the  ratio 
5:6:7:3. 

18.  The  sides  of  a  triangle  are  in  the  ratio  5:8:9,  and  its 
perimeter  is  6'  5".     Find  the  sides. 

19.  Divide  the  circumference  of  a  circle  whose  diameter  is 
16"  into  three  parts  in  the  ratio  3:5:7. 

20.  Five  angles  about  a  point  on  one  side  of  a  straight  line 
are  in  the  ratio  1:2:3:4:5.     Find  them. 


Proportion 

98  Proportion.  A  proportion  is  an  equation  in  which  the  two 
members  are  ratios.  For  example :  y^  =  ii  is  a  proportion,  and 
may  be  read  8  is  to  12  as  16  is  to  24.  The  first  and  fourth  terms 
of  a  proportion  are  called  the  extremes,  and  the  second  and  third 
are  called  the  means.  In  the  proportion  -^  =  ^^,  8  and  24  are 
the  extremes,  and  12  and  16,  the  means. 

Example:    Solve    1^  =  9 

15  =  4x   (clearing  of  fractions.) 
x  =  3f 

Check:    A=— 
^^      9 

5    _    5 
T2— T2 


PROPORTION 

Exercise  62* 

Solve  and  check : 

1.  ^  =  i? 
25  14 

6. 

11  18 

12  X 

2.  ^  =  i? 

7   17 

6. 

125  206 
X   305 

3   ^-^ 
^-   x'll 

7. 

144  3x 
195  25 

4  ^  =  ^ 

9   14 

8. 

X   3j 
24  41 

87 


9.    The  ratio  of  x+1  to  9  is  equal  to  the  ratio  of  x+5  to  15. 
Find  x. 

10.  The  ratio  of  the  complement  of  an  angle  to  the  angle  is 
equal  to  the  ratio  y.     Find  the  angle. 

11.  The  ratio  of  the  supplement  of  an  angle  to  the  angle  is 
equal  to  the  ratio  y^ .     Find  the  angle. 

12.  The  ratio  of  an  angle  to  84°  is  equal  to  the  ratio  of  its 
complement  to  96°.     Find  the  angle. 

13.  One  number  is  5  larger  than  another,  and  the  ratio  of  the 
larger  to  the  smaller  is  equal  tof  .     Find  the  two  numbers. 

14.  The  length  of  a  rectangle  is  6  more  than  its  width,  and 
the  ratio  of  the  length  to  the  width  is  ^.  Find  the  dimensions 
of  the  rectangle. 

16.  Two  numbers  are  in  the  ratio  f .  If  2  is  added  to  the 
smaller,  the  ratio  of  that  number  to  the  larger  is  f .  Find  the 
numbers.     (See  Example,  Art.  97.) 

16.  If  the  scale  of  a  drawing  is  J"  to  1',  how  long  should  a 
line  be  made  in  the  draving  to  represent  32'? 


88  RATIO,    PROPORTION,   AND   VARIATION 

17.  If  the  scale  of  a  drawing  is  f "  to  1',  how  long  should  a 
line  be  made  to  represent  10"? 

18.  If  the  scale  of  a  drawing  is  ij"  to  1',  what  line  would  be 
represented  by  a  line  3|"  on  the  drawing? 

19.  If  a  drawing  is  to  be  reduced  to  f  its  size,  what  would  be 
the  length  on  the  new  drawing,  of  a  dimension  3^"  on  the 
original  drawing? 

20.  If  a  dimension  line  f "  on  a  drawing  represents  a  line  4j" 
long,  what  is  the  scale  of  the  drawing? 

99  It  is  often  necessary  in  shop  practice  to  express  a  fraction  or 
decimal  in  halves,  fourths,  eighths,  sixteenths,  etc.  A  proportion 
is  a  convenient  means  of  changing  to  these  denominators. 

Example :     How  many  g^'s  in  y^. 

3^  2^  s  in  Y  5^. 


Let  X  =  number  of  -oV's  in  — — 


32     15 
15x  =  352 
x  =  23y5^,  approximately  23^. 

Exercise  63 

1.  How  many  J's  in  -^q? 

2.  How  many  y^'^  i^  i'^ 

3.  How  many  -^'s  in  .3? 

4.  Reduce  1.312  to  eighths. 

5.  Reduce  1^%-  to  sixty-fourths. 


PROPORTION  89 


X         4 

100  Example:     1.     Solve   — rT  =  "r 

x-j-l      5 


Check: 


4^4 
4+l~  5 

5       5 


5x  =  4x+4  iL.  C.  D.  is5(x+l)> 
X  =  4     Why? 


X  1 


Example:    2.     Solve  ^,  ^  ,,  =^ 

o(x  — 1)       6 

2x  =  x-l  |l.  C.  D.  is6(x-l)| 
x=-l     Why? 


Check; 


-1  1 


3(-l-l)      6 

-6      6 

6      6 


Example:    3.     Solve 


x+l     x-3 


::heck: 


x+2     x-4 
x2-3x-4  =  x2-x-6  I  L.  C.  D.  is  (x+2)  (x-4) 
—  2x=-2     Why? 
X  =      1      Why? 

1  +  1  ^  1-3 

1  +  2       1-4 

3-3 

1  =  1 
3       3 


90                         RATIO,    PROPORTION,  AND   VARIATION 

Exercise  64 
Solve  and  check: 

1      _^=1  7      -^ =  i 

x-1     4  3(5x-6)     9 

_x ^4  7(3x-7)^23 

^*     3(x-l)     9  4(x+3)      12 

y       ^  1  2^3 

^*     5(y+2)"l0  3x+l     5x+2 

4.     ^-±?4  10.          '              ' 


x-5     6  4x-3    3x+4 

X         2  ^^      x+5     x+25 


3x+l     7  x-4     x-2 

2y+3^3  5x-7^10x+ll 

3y+7    4  3x-5       6x+  7 


13. 


2x-3  3x 


2(x-3)     3x-4 

14.  The  ratio  of  an  angle  to  its  supplement  is  J.  Find  the 
angle. 

15.  The  ratio  of  an  angle  to  its  complement  is  y.  Find  the 
angle. 

16.  The  ratio  of  the  supplement  of  an  angle  to  the  comple- 
ment is  f .     Find  the  angle. 

17.  If  an  angle  is  increased  by  3°  and  its  complement  de- 
creased by  13°,  the  ratio  of  the  two  angles  will  then  be  -|.  Find 
the  original  angle. 

18.  The  base  of  one  rectangle  is  3  less  than  the  base  of 
another.  The  altitude  of  the  first  is  3,  and  that  of  the  second 
is  5.  The  ratio  of  the  areas  is  f .  Find  the  bases  of  the 
two  rectangles. 

19.  The  ratio  of  3°  to  the  complement  of  an  angle  is  equal  to 
the  ratio  of  21°  to  the  supplement  of  the  same  angle.  Find  the 
angle. 


DIRECT   VARIATION  91 

20.  Find  three  consecutive  numbers  such  that  the  ratio  of 
the  first  to  the  second  is  equal  to  the  ratio  of  5  times  the  third 
to  5  times  the  first  plus  16. 

Variation 


101  Direct  Proportion:  If  a  train  travels  120  miles  in  3  hours,  it 
would  travel  240  miles  in  6  hours,  -f  is  the  ratio  of  the  two 
times,  and  -J^f  ^  is  the  ratio  of  the  two  distances,  taken  in  the  same 
order.  Both  ratios  reduce  to  J  and  therefore  the  problem  may 
be  expressed  by  the  proportion,  -f  =  i^%.  An  increase  in  time 
produces  an  increase  in  distance. 

If  the  train  travels  120  miles  in  3  hours,  it  would  travel  80 
miles  in  2  hours  because  |-  =  ^-^-^ .  A  decrease  in  time  produces  a 
decrease  in  distance. 

When  two  quantities  are  so  related  that  an  increase  or  de- 
crease in  one  produces  the  same  kind  of  a  change  in  the  other, 
one  is  said  to  be  directly  proportional  to  the  other,  or  to  vary 
directly  as  the  other. 

Example:  If  a  piece  of  steel  3  yds.  long  weighs  270  lbs.,  how 
much  will  a  piece  5  yds.  long  weigh? 

Let  X  =  weight  of  the  5-yd.  piece. 

3     270 

-  =  — —  (the  weight  is  directly  proportional  to  the  length.) 
5       X 

x=   ? 

Exercise  65 

1.  If  60  cu.  in.  of  gold  weighs  42  lbs.,  how  much  will  35  cu. 
in.  weigh? 

2.  If  the  interest  on  a  certain  sum  of  money  is  $84.20  for  5 
yrs.,  what  would  be  the  interest  for  8 J  yrs.? 

3.  If  a  section  of  I-beam  10  yds.  long  weighs  960  lbs.,  how 
long  is  a  piece  of  the  same  material  weighing  1280  lbs.? 


92  RATIO,    PROPORTION,    AND    VARIATION 

4.  An  engine  running  at  320  revolutions  per  minute 
(R.P.M.)  develops  8^  horsepower.  How  many  horsepower 
would  it  develop  at  365  R.  P.  M.? 

5.  At  40  lbs.  pressure  per  square  inch,  a  given  pipe  dis- 
charges 180  gallons  per  minute.  How  many  gallons  per  minute 
would  be  discharged  at  55  lbs.  pressure? 

6.  What  will  be  the  resistance  of  a  mile  of  wire  if  the 
resistance  of  500  yds.  of  the  same  wire  is  .65  ohms? 

7.  A  steam  shovel  can  handle  900  cu.  yds.  of  material  in 
8  hrs.  At  the  same  rate  how  many  cu.  yds.  can  be  handled 
in  7  hrs.? 

8.  A  12  pitch  gear  10"  in  diameter  has  120  teeth.  How 
many  teeth  would  a  6"  gear  with  the  same  pitch  have? 

9.  An  engine  running  at  185  R.  P.  M.  drives  a  line  shaft 
at  210  R.  P.  M.  At  what  R.  P.  M.  should  an  engine  run  to 
give  the  line  shaft  a  speed  of  240  R.  P.  M.? 

10.  If  a  machine  can  finish  65  pieces  in  75  minutes,  how 
long  will  it  take  it  to  finish  104  pieces?  * 

102  Inverse  Proportion:  K  a  train  travels  a  given  distance  in 
4  hrs.  at  the  rate  of  40  miles  per  hour,  it  would  take  8  hrs. 
to  travel  the  same  distance  if  the  rate  were  20  miles  per  hour. 
■^  is  the  ratio  of  the  two  times,  and  -|^  is  the  ratio  of  the  two 
rates,  taken  in  the  same  order.  |-  =  J  but  ■f§  =  |-.  Therefore 
the  problem  may  be  expressed  as  a  proportion  if  one  ratio  is 
first  inverted.  ^  =  ^%  or  |-  =  -f^.  An  increase  in  time  produces 
a  decrease  in  rate. 

If  a  train  travels  a  given  distance  in  4  hours  at  the  rate  of 
40  miles  per  hour,  it  would  take  2  hours  to  travel  the  same 
distance  if  the  rate  were  80  miles  per  hour,  because  |^  =  -|^  or 
1^  =  1^-0^.     A  decrease  in  time  produces  an  increase  in  rate. 


INVERSE    VARIATION  93 

When  two  quantities  are  so  related  that  an  increase  or  a 
decrease  in  one  produces  the  opposite  kind  of  a  change  in  the 
other,  one  is  said  to  be  inversely  proportional  to  the  other,  or 
to  vary  inversely  as  the  other. 

Example:  If  6  men  can  do  a  piece  of  work  in  10  days, 
how  long  will  it  take  5  men  to  do  it? 

Let  X  =  time  it  will  take  5  men 

6 X^  (the  number  of  men  is  inversely  proportional  to 

^  ~  jQ        the  number  of  days.) 

x  =  ? 

Exercise  66 

1.  A  train  traveling  at  the  rate  of  50  miles  per  hour  covers 
a  distance  in  5  hrs.  How  long  would  it  take  to  cover  the  same 
distance  if  it  traveled  at  40  miles  per  hour? 

2.  A  man  walking  at  4  miles  per  hour  can  travel  a  dis- 
tance-in 3  hrs.  At  what  rate  would  he  have  to  walk  to  cover 
it  in  2  hrs.? 

3.  If  40  men  can  do  a  piece  of  work  in  10  days,  how  long 
will  it  take  25  men  to  do  it? 

4.  12  men  can  do  a  piece  of  work  in  28  days.  How  many 
men  could  do  it  in  84  days? 

5.  The  number  of  posts  required  for  a  fence  is  42  when 
they  are  placed  18  ft.  apart.  How  many  would  be  needed 
if  they  were  placed  14  ft.  apart? 

6.  One  investment  of  $6,000  at  3|%  yields  the  same 
income  as  another  at  3%.  What  is  the  amount  of  the  second 
investment? 

7.  A  man  has  two  investments,  one  of  $15,900,  and  the 
other  $21,200.  The  first  is  invested  at  6%.  At  what  rate 
must  the  other  be  invested  to  produce  the  same  income  as 
the  first? 


94  RATIO,    PROPORTION,    AND   VARIATION 

8.  A  man  planned  to  use  36  posts  spaced  9  ft.  apart  in 
building  a  fence.  His  order  was  6  posts  short.  How  far 
apart  should  he  place  them? 

Exercise  67.     (Review) 

1.  The  circumference  of  a  circle  is  directly  proportional 
to  its  diameter.  If  the  circumference  of  a  circle  whose  diam- 
eter is  6"  is  18.8496",  what  is  the  circumference  of  a  circle 
whose  diameter  is  4"? 

2.  If  the  circumference  of  a  circle  whose  diameter  is  5" 
is  15.708",  what  is  the  diameter  of  a  circle  with  a  circumference 
of  28.2744"? 

3.  The  area  of  a  circle  varies  directly  as  the  square  of  its 
diameter.     If  the  area  of  a  2"  circle  is  12.5664  sq.  in.,  find  the 

12.5664     4 
area  of  a  4"  circle.     (Suggestion :  =  — ) 

4.  The  volume  of  a  quantity  of  gas  varies  inversely  as 
the  pressure  when  the  temperature  is  constant.  If  the  volume 
of  a  gas  is  600  cubic  centimeters  (c.  c.)  when  the  pressure  is 
60  grams  per  square  centimeter,  find  the  pressure  when  the 
volume  is  150  c.  c. 

5.  A  quantity  of  gas  measures  423  c.  c.  under  a  pressure 
of  815  millimeters  (m.  m.).  What  will  it  measure  under  760 
m.  m.? 

6.  The  volume  of  a  cube  varies  directly  as  the  cube  of  the 
edge.  If  the  volume  of  an  11"  cube  is  1331  cu.  in.,  what  is 
the  volume  of  a  7"  cube?     (See  suggestion,  problem  3.) 

7.  The  volume  of  a  sphere  is  directly  proportional  to  the 
cube  of  its  diameter.  Find  the  volume  of  a  6"  sphere  if  a 
10"  sphere  contains  523.6  cu.  in. 

8.  The  volume  of  a  quantity  of  gas  varies  directly  as  the 
absolute  temperature  when  the  pressure  is  constant.     If  a 


REVIEW  95 

quantity  of  gas  occupies  3.25  cu.  ft.  when  the  absolute  tem- 
perature is  287°,  what  will  be  its  volume  at  329°? 

9.  The  velocity  of  a  falling  body  varies  directly  as  the 
time  of  falHng.  If  the  velocity  acquired  in  4  seconds  is  128.8 
ft.  per  sec,  what  would  be  the  velocity  acquired  in  7  seconds? 

10.  The  weight  of  a  disk  of  copper  cut  from  a  sheet  of 
uniform  thickness  varies  as  the  square  of  the  diameter.  Find 
the  weight  of  a  circular  piece  of  copper  12"  in  diameter  if  one 
7"  in  diameter  weighs  4.42  oz. 

11.  A  wheel  28"  in  diameter  makes  42  revolutions  in  going 
a  given  distance.  How  many  revolutions  would  a  48"  wheel 
make  in  going  the  same  distance? 

12.  If  3  men  can  build  91  rods  of  fence  in  a  certain  time, 
how  much  could  7  men  build  in  the  same  time? 

13.  If  25  men  can  do  a  piece  of  work  in  30  days,  how  long 
would  it  take  27  men  to  do  the  same  work? 

14.  If  the  pressure  on  230  c.  c.  of  nitrogen  is  changed 
from  760  m.  m.  to  665  m.  m.,  what  will  be  its  new  volume? 

15.  The  absolute  temperature  of  730  c.  c.  of  hydrogen  is 
changed  from  353°  to  273°.     What  is  its  new  volume? 

16.  If  the  circumference  of  a  circle  3 J"  in  diameter  is 
10.9956,  what  is  the  diameter  of  a  circle  whose  circumference 
is  23.562? 

17.  A  sum  of  money  earns  S1750  in  3j  yrs.  How  long  will 
it  take  it  to  earn  $2750? 

18.  An  investment  of  $1125  at  5%  earns  the  same  amount 
as  another  of  $1250.    What  is  the  rate  of  the  second  investment? 

19.  If  an  investment  at  2j%  produces  an  income  of  $400, 
what  would  it  produce  if  invested  at  3f  %? 

20.  The  diameter  of  a  sphere  which  contains  47.71305  cu. 
in.  is  4|".     What  will  a  sphere  contain  whose  diameter  is  3"? 


CHAPTER  VI . 

PULLEYS,  GEARS,  AND  SPEED 

103  An  important  problem  in  the  running  of  lathes  is  the  cal- 
culation of  the  speed  at  which  the  work  should  be  turned,  in 
order  to  complete  the  work  in  the  shortest  time  possible,  with- 
out injury  to  the  work  or  the  tools  used.  Similar  problems 
arise  in  the  use  of  fly-wheels,  emery  wheels,  grindstones,  etc. 

104  Rim  Speed:  When  the  work  in  a  lathe  is  turned  through 
one  complete  revolution,  a  point  upon  the  surface  of  the  work 
travels  a  distance  equal  to  the  circumference  of  the  work.  In 
one  minute,  it  would  travel  a  distance  equal  to  the  circum- 
ference of  the  work  multiplied  by  the  number  of  revolutions 
per  minute  (R.  P.  M.). 

The  distance  in  feet  traveled  by  a  point  on  the  circumference 
of  a  wheel  in  one  minute  is  called  Rim  Speed  or  Surface  Speed. 

105  RULE:     To  find  the  rim  speed,  multiply  the  circumference  of  the 

revolving  object  by  the  number  of  revolutions  per  minute  (R.  P. 
M.)f  and  express  the  result  in  feet. 

Example  1:    The  diameter  of  a  wheel  is  2".     If  it  makes 
2500  R.  P.  M.,  what  is  the  rim  speed? 

C  =  ;r.D  =  3.1416 -2  =  6.2832". 

6.2832 

^^     =  .5236  (circumference  in  feet.) 

.5236X2500=  1309,  rim  speed. 

Example  2:    The  surface  speed  of  a  wheel  is  3000.     If  the 
diameter  is  4",  what  is  its  R.  P.  M.? 

3.1416.4  =  12.5664" 

12.5664 

^fy      =  1.0472  (circumference  in  feet.) 

96 


RIM  SPEED  97 

Letx  =  R.  P.  M. 

then  1.0472x  =  3000 

x  =  2865,  R.  P.  M. 

Example  3 :     What  is  the  diameter  of  a  wheel  if  its  R.  P.  M. 
is  2500  and  its  surface  speed  is  1500  ft.  per  minute? 

Let  x  =  diameter  of  the  wheel. 

Then  3. 1416x  =  circumference  of  the  wheel. 
3.1416X.  2500  =  1500 
7854x=1500 

X  =  .19,    diameter  in  feet. 
.19  •  12  =  2.28  diameter  in  inches. 

Exercise  68 

1.  What  would  be  the  rim  speed  of  a  12'  fly  wheel  running 
at  75  R.  P.  M.? 

2.  An  emery  wheel  15"  in  diameter  runs  at  1400  R.  P.  M. 
Find  the  surface  speed. 

3.  A  pulley  5i"  in  diameter  runs  at  1250  R.  P.  M.  What 
is  its  rim  speed? 

4.  A  12"  circular  saw  runs  at  2450  R.  P.  M.  What  is 
its  cutting  speed  (rim  speed)? 

5.  A  10"  emery  wheel  has  a  rim  speed  of  5000  ft.  per 
minute.     How  many  R.  P.  M.  does  it  make? 

6.  A  grindstone  will  stand  a  surface  speed  of  800  ft.  per 
minute.  At  how  many  R.  P.  M.  can  it  run  if  its  diameter 
is  4'  8"? 

7.  At  how  many  R.  P.  M.  should  a  9|"  shaft  be  turned 
in  a  lathe  to  give  a  cutting  speed  of  60  ft.  per  minute? 

8.  A  fly  wheel  having  a  rim  speed  of  a  mile  a  minute 
runs  at  120  R.  P.  M.     What  is  its  diameter? 


98  PULLEYS,    GEARS,    AND   SPEED 

9.     An  emery  wheel  runs  at  950  R.  P.  M.      If  its  surface 
speed  is  5500  ft.  per  minute,  what  is  its  diameter? 

10.  A  line  shaft  runs  at  186  R.  P.  M.  A  pulley  on  this 
shaft  has  a  riih  speed  of  1350  ft.  per  minute.  What  is  the 
diameter  of  the  pulley? 

11.  The  splicing  of  a  belt  connecting  two  equal  pulleys 
travels  through  the  air  at  the  rate  of  2000  ft.  per  minute. 
At  what  speed  must  the  pulleys  run  if  they  are  20"  in  diameter? 

12.  A  band  saw  runs  over  two  pulleys  each  32"  in  diameter. 
If  the  band  saw  is  16'  long,  and  the  speed  of  the  wheels  600 
R.  P.  M.,  what  is  the  cutting  speed  of  the  band  saw? 

Pulleys 


Fig.  61.  Pulleys 


106  When  two  pulleys  are  connected  by  a  belt,  the  rim  speeds 
of  the  two  pulleys  must  be  the  same  if  there  is  no  slipping  of 
the  belt.  Suppose  pulley  I  (Fig.  61)  is  6"  in  diameter  and 
pulley  II  is  12".  The  circumference  of  I  is  J  as  large  as  the 
circumference  of  II,  and  therefore  I  will  revolve  twice  while 
II  revolves  once.  In  other  words,  the  ratio  of  the  diameter 
of  I  to  the  the  diameter  of  II  is  J,  while  the  ratio  of  the  R.  P.  M. 
of  I  to  the  R.  P.  M.  of  II  is  y.  This  may  be  expressed  as  a 
proportion  if  one  ratio  is  inverted  and  therefore : 


PULLEYS  99 

107  When  two  pulleys  are  connected  hy  a  belt,  the  size  of  the  pulley 
varies  inversely  as  its  R.  P.  M. 

Example:  One  of  two  pulleys  connected  by  a  belt  is  12" 
in  diameter,  and  its  R.  P.  M.  is  400.  What  is  the  R.  P.  M. 
of  the  other  pulley  if  it  is  3"  in  diameter? 

Let  x  =  R.  P.  M.  of  the  second  pulley. 

12       X 

—  = (the  size  varies  inversely  as  the  R.  P.  M.) 

3      400 

x=1600.  R.  P.  M. 


Exercise  69 

1.  A  12"  pulley  running  at  200  R.  P.  M.  drives  an  8" 
pulley.     Find  the  R.  P.  M.  of  the  8"  pulley. 

2.  A  14"  pulley  drives  a  26"  pulley  at  175  R.  P.  M.  What 
is  the  R.  P.  M.  of  the  14"  pulley? 

3.  A  30"  pulley  running  240  R.  P.  M.  is  belted  to  a  12" 
pulley.     Find  the  R.  P.  M.  of  the  12"  pulley. 

4.  A  pulley  on  a  shaft  running  at  120  R.  P.  M.  drives  a 
24"  pulley  at  200  R.  P.  M.  What  is  the  diameter  of  the  pulley 
on  the  shaft? 

Lineshaft:  The  line  shaft  is  the  main  shaft  which  drives 
the  machinery  of  a  shop  by  means  of  pulleys  and  belts. 

Counter  Sfiaft:  A  counter  shaft  is  an  auxiliary  shaft  placed 
between  the  line  shaft  and  a  machine  to  permit  a  convenient 
location  of  the  machine. 

5.  A  line  shaft  runs  at  250  R.  P.  M.  Determine  the  size 
of  the  pulley  on  the  line  shaft  in  order  to  run  a  6"  pulley  on 
a  machine  at  1550  R.  P.  M. 


100  PULLEYS,    GEARS,    AND   SPEED 

6.  It  is  found  necessary  to  run  a  counter  shaft  at  310 
R.  P.  M.  If  driven  by  an  18"  pulley  running  at  175  R.  P.  M., 
what  must  be  the  diameter  of  the  pulley  on  the  counter  shaft? 

7.  A  counter  shaft  for  a  grinder  is  to  be  driven  at  375 
R.  P.  M.  by  a  line  shaft  that  runs  at  210  R.  P.  M.  If  the 
pulley  on  the  counter  shaft  is  12"  in  diameter,  what  size  pulley 
should  be  put  on  the  line  shaft? 

8.  A  motor  running  at  875  R.  P.  M.  has  a  10|"  driving 
pulley.  If  the  motor  drives  a  line  shaft  at  180  R.  P.  M., 
what  must  be  the  size  of  the  line  shaft  pulley? 

9.  The  diameters  of  two  pulleys  connected  by  a  belt  are 
in  the  ratio  f .  If  the  R.  P.  M.  of  the  larger  pulley  is  966, 
what  is  the  R.  P.  M.  of  the  smaller? 


Fig.  62 


10.  Pulley  I  is  belted  to  II,  and  III  to  IV  (Fig.  62).  II  and 
III  are  on  the  same  shaft.  If  the  diameter  of  I  is  18"  and  its 
R.  P.  M.  is  240,  find  the  R.  P.  M.  of  II  if  its  diameter  is  8". 
Find  the  R.  P.  M.  of  IV  if  it  has  a  diameter  of  6",  and  III  has 
one  of  20". 


PULLEYS 

Step-Cone  Pulleys 


101 


Fig.  63,  Step-Cone  Pulleys 

108  To  secure  different  speeds  on  the  same  machine,  step-cone 
pulleys  (Fig.  63)  are  used  on  both  the  driving  shaft  and  the 
driven  shaft.  The  large  step  of  the  driving  pulle>  may  be 
belted  to  the  small  one  of  the  driven  for  high  speed,  the  medium 
one  to  the  medium  one  for  middle  speed,  and  the  small  one  to 
the  large  one  for  low  speed. 

Example:  A  step-cone  pulley  having  diameters  11",  8 J", 
and  6",  running  at  120  R.  P.  M.,  drives  a  step-cone  pulley 
having  diameters  4",  6^",  and  9".     Find  the  three  speeds. 


Let  x  =  R.  P.  M.  at  high  speed. 

Why? 

1320  =  4x 

X  =  330,  R.  P.  M.  at  high  speed. 


T'>-T=i|-0 


102  PULLEYS,    GEARS,   AND   SPEED 

Let  y  =  R.  P.  M.  at  middle  speed. 

Then -4=-^  Whv? 
6  J     120 

1020  =  6^y. 

y  =  157  — ,  R.  p.  M.  at  middle  speed. 

Let  z  =  R.  P.  M.  at  low  speed. 

6       z 

Then  -= — •  Why? 

9     120 

720  =  9z. 

Z  =  80  R.  P.  M.  at  low  speed. 


Exercise  70 


1 

it 

"'"""III"' 

o 

r 

J* 

t 

Fig    64 


Fig.  65 


1.  The  steps  of  a  pair  of  cone  pulleys  are  7",  5",  3",  and 
4",  6",  8"  in  diameter  (Fig.  64).  If  the  lower  pulley  has  a 
speed  of  1050  R.  P.  M.,  find  the  three  speeds  of  the  upper 
pulley. 

2.  The  diameters  of  the  steps  of  a  step-cone  pulley  on  a 
machine  are  10",  8|"  and  7",  and  the  corresponding  counter 
shaft  diameters  are  5^'',  7",  and  8j".  Find  the  speed  for  each 
step  on  the  machine  if  the  counter  shaft  runs  at  1190  R.  P.  M. 


GEARS  103 

3.  The  steps  of  the  cone  pulley  on  a  wood-turning  lathe 
are  7|",  5f ",  and  4".  The  corresponding  diameters  of  the 
driving  pulley  on  the  motor  are  2f ",  4|",  and  6j".  Find  the 
three  speeds  on  the  lathe  if  the  motor  speed  is  1165  R.  P.  M. 

4.  The  smallest  steps  on  a  pair  of  cone  pulleys  are  2|" 
and  2f ".  The  increase  in  diameter  of  each  succeeding  step  is 
1^'  (Fig.  65).  The  first  pulley  has  a  speed  of  1000  R.  P.  M. 
Find  the  three  speeds  of  the  second  pulley. 


Gears 


Fig.  66.    Gears 

109  In  machines  where  absolute  accuracy  in  the  speed  of  the 
work  is  required,  gears  are  used  instead  of  belts  to  eliminate 
slipping.  When  two  gears  are  meshed  as  in  Fig.  66,  it  is  evi- 
dent that  their  rim  speeds  are  the  same.  Sizes  of  gears  are 
measured  by  the  number  of  teeth  rather  than  their  diameters. 
Suppose  a  48-tooth  gear  drives  one  with  24  teeth.  The  smaller 
one  will  revolve  twice,  while  the  larger  one  revolves  once. 
The  ratio  of  the  numbers  of  teeth  is  f,  while  the  ratio  of  the 
speeds  is  -J.     Therefore : 

110  When  one  gear  drives  another,  the  speed  is  inversely  propor- 
tional  to  the  number  of  teeth. 


104 


PULLEYS,    GEARS,    AND   SPEED 

Exercise  71 


1.  A  38-tooth  gear  is  driving  one  with  72  teeth.  If  the 
first  gear  runs  at  360  R.  P.  M.,  what  is  the  speed  of  the  second 
gear? 

2.  A  14-tooth  gear  running  at  195  R.  f*.  M.  is  to  drive 
another  gear  at  105  R.  P.  M.  What  must  be  the  number  of 
teeth  in  the  second  gear? 

3.  Two  gears  are  to  have  a  speed  ratio  of  3  to  4.  If  the 
first  gear  has  36  teeth,  how  many  will  the  second  have? 

4.  The  ratio  of  the  numbers  of  teeth  in  two  gears  is  y. 
The  R.  P.  M.  of  the  first  is  350.  What  is  the  speed  of  the 
second? 


^6T 


Fig.  67 


6.  In  Fig.  67  gear  I  has  72  teeth,  II  has  40,  III  has  56, 
and  IV  has  32.  The  R.  P.  M.  of  gear  I  is  60.  Find  the 
R.  P.  M.  of  II.  If  gear  III  is  on  the  same  shaft  as  II,  find 
the  R.  P.  M.  of  IV. 


REVIEW   PROBLEMS 

Exercise  72.     (Review) 


lO" 


Fig.  68 

1.  The  gear  with  72  teeth  has  a  speed  of  35  R.  P.  M. 
Find  the  speed  of  the  32-tooth  gear.     (Fig.  68.) 

2.  If  the  32-tooth  gear  (Fig.  68)  is  to  be  replaced  by  one 
which  is  to  have  a  speed  of  280  R.  P  M.,  what  size  gear  must 
be  used? 


Fig.  69 


3.  In  Fig.  69  what  must  be  the  size  of  the  line  shaft 
pulley  (I)  to  run  the  emery  wheel  (V)  at  1215  R.  P.  M.,  if  the 
R.  P.  M.  of  the  line  shaft  is  150? 


106  PULLEYS,    GEARS,    AND   SPEED 

4.  What  would  be  the  R.  P.  M.  of  the  emery  wheel  (V), 
Fig.  69,  if  the  line  shaft  pulley  (I)  is  replaced  by  a  48"  pulley? 

6.  Find  the  grinding  speed  of  the  emery  wheel  in  problem 
4,  if  its  diameter  is  12". 

6.  A  wood-turning  lathe  is  driven  by  a  motor  running  at 
1200  R.  P.  M.  The  smallest  step  of  the  cone  pulley  on  the 
motor  shaft  is  2"  in  diameter,  and  its  mate  on  the  lathe  is  7". 
All  increases  in  the  diameters  of  succeeding  steps  are  2".  If 
the  work  being  turned  is  3"  in  diameter,  find  the  cutting  speed 
on  high  speed. 

7.  Find  the  cutting  speed  in  problem  6  on  middle  speed. 

8.  Find  the  cutting  speed  in  problem  6  on  low  speed. 


CHAPTER  VII 

SQUARES  AND  SQUARE  ROOTS 

111  Square  of  a  binomial:  A  few  kinds  of  multiplication  prob- 
lems are  used  so  often  that  it  is  a  saving  of  time  to  be  able  to 
write  the  result  without  performing  the  actual  multiplication. 
One  of  these  is  the  square  of  a  binomial. 

Find  the  value  of  the  following  by  multiplying,  and  write 
the  results  as  in  part  1: 

1.  (a+3)2  =  a2H-6a4-9.  3.     (m+n)2  = 

2.  (b+5)2=  4.     (x+7y)2  = 
In  each  result,  observe  the  following: 

I.     There  are  3  terms  in  the  result. 

II.  The  first  term  of  the  result  is  the  square  of  the  first 
term  of  the  binomial,  and  the  third  term  of  the  result  is  the 
square  of  the  second  term  of  the  binomial. 

III.     The  second  term  of  the  result  is  2  times  the  product 
of  the  two  terms  of  the  binomial. 

Find  the  value  of  the  following  by  actual  multiplication  and 
write  the  results  as  in  part  1 : 

1.  (a-3)2  =  a2-6a+9.  3.     (-m+n)2  = 

2.  (b-10)2=  4.     (-x-7y)2  = 

In  each  result  observe  that  the  same  facts  hold  true  as  in 
the  preceding  case,  and  that  the  law  of  signs  for  multiplication 
must  be  used. 

112  RULE:    To  square  a  binomial,  square  the   first   term,  take   2 

times  the  product  of  the  two  terms,  square  the  second  term, 
and  write  the  result  as  a  trinomial. 

107 


108  SQUARES  AND  SQUARE  ROOTS 

Example:     (2a-3bx)2  =  (+2a)2+2(+2a)(-3bx)  +  (-3bx)^ 
=  (-f4a2)  +  ( -  12abx)  +  (+9bV) 
=  4a2-12abx+9b2x2. 


Exercise  73 

Wri 

te  the  results  without  written  multiplication: 

1. 

(a+l)2 

16. 

(a2+b2)2 

2. 

(t+u)^ 

17. 

(2m2-3n)2 

3. 

(d-4)2 

18. 

(4t3-3u2)2 

4. 

(x-y)^ 

19. 

(a4+4a)2 

6. 

(2a+b)2 

20. 

(7-3m2)2 

6. 

(3x-5)2 

21. 

(m-J)2 

7. 

(a-3b)2 

22. 

(y+*)^ 

8. 

(x+4y)2 

23. 

(2x-i)2 

9. 

(2m+3n)2 

24. 

(3m+f)2 

10. 

(5t-4u)2 

26. 

(4x+iy)2 

11. 

(6ab-5xy)2 

26. 

(|x^-fy)^ 

12. 

(5ab+4bx)2 

27. 

(Ift2-fu3)2 

13. 

(m2+5)2 

28. 

(ff+sO^ 

14. 

(x2-8)2 

29. 

(2.3  l-5.1m)2 

15. 

(a2-2)2 

30. 

(.3125m3n+3|mn3)2 

31.    Square  32  mentally. 
Suggestion  322=  (30+2)2 

=  900+120+4  =  1024 

Square  the  following  mentally: 

32.  21  36.  34 

33.  22  36.  37 

34.  29  (Suggestion  29  =  30-1)  37.  49 


38. 

19 

39. 

35 

40. 

43 

SQUARE   ROOT   OF   MONOMIALS  109 

SQUARE  ROOT 

Square  Root  of  Monomials 

113  Square  Root:  Problems  often  arise  in  which  the  reverse  of 
squaring  is  necessary.  For  example:  what  must  be  the  side 
of  a  square  whose  area  is  25  sq.  in.?  The  side  must  be  such 
that,  if  multiplied  by  itself,  the  result  will  be  25.  It  is  evident 
that  5  is  the  side  of  the  square  since  5^  =  25. 

The  square  root  of  a  number  is  a  number  which  if  squared,  will 
produce  the  given  number. 

Finding  such  a  number  is  called  extracting  square  root,  and 
the  operation  is  indicated  by  the  radical  sign,  V 

( 1 4)2 1 16  }  .-.  V"l6  =  +4,  or  -  4,  written  +4. 

(-3a)2  =  9a2\       .-^-^     ,  „ 
(+3a)2  =  9a2/-'-^^^  =±^^' 


(Ila2b4)2  =  121a%8.-. 

V121a%«  = 

^lla^b^. 

Exercise  74 

Find  the  square  root  of: 

1.    81 

4. 

144m2n2 

2.     121 

5. 

25xV 

3.    4a2 

Find  the  value  of: 

8. 
9. 

6.     V49xy 

V  196xV' 

7.     V64a2b^ 

V256ci«d«e^ 

10.     V400a2b4c8 

The  square  root  of  a  negative  number  cannot  be  found  since,  by  the 
law  of  signs  for  multiplication,  the  square  of  either  a  positive  or  a  negative 
number  is  positive. 


no 


SQUARES  AND  SQUARE  ROOTS 

Square  Root  of  Trinomials 


114  (a+b)2  =  a2+2ab+b2   \  .    .  ,  ,  ^  ,    ,  , ,     ,      ,  ,  , 

(_a_b)2  =  a2+2ab+b/-- ^^ +2ab+b2=+aH-b,or-a-b, 

written  +  (a +b). 
(a-b)2  =  a2-2ab+b2    \  . 


(_a+b)2  =  a2-2ab+bV  •*•  Va2-2ab+b2= +a-b,or-aH-b, 

written  +(a  — b). 

115  Trinomial  Square:  A  trinomial  in  which  two  of  the  terms 
are  squares  and  positive,  and  the  other  term  is  2  times  the 
product  of  the  square  roots  of  those  terms,  is  called  a  trinomial 
square,  and  is  the  square  of  a  binomial. 


Exercise  75 
Select  the  trinomial  squares  in  the  following: 

11.  64a^-176a+121 

12.  49m4n2+112m2+8n4 

13.  x2+x+i 

14.  m2+fm+^ 


1.  x2+2xy+y2 

2.  m^— 4m+4 

3.  m2-4m+6 

4.  a2-4a-4 

5.  x2-6xy4-9y2 

6.  4t2+6tu+9u2 

7.  16x2+25y2 

8.  169m6-26m3n+n2 

9.  25x2+16y2-40xy 
10.  49V-70xyz-25z2 


15.  x2+|x+| 

16.  4a2+ab+YVb' 

17.  9m2-24mn  +  16 

18.  x2  +  |x  +  y% 

19.  y^+fy+A 

20.  t^-it-aV^e 


SQUARE   ROOT   OF   TERMINALS  111 


116  By  Art.  114,    Va2+2ab+b2= -f(a+b). 

Va2-2ab+b2=-|-(a-b). 

Observe  the  following  facts  in  each  result: 

I.    The  two  terms  of  the  binomial  are  the  square  roots  of 
the  two  terms  of  the  trinomial  which  are  squares. 

II.  If  the  sign  of  the  other  term  of  the  trinomial  is  plus, 
the  terms  of  the  binomial  have  like  signs,  and  if  it  is  minus, 
the  terms  of  the  binomial  have  unlike  signs. 

117  RULE:    To  find  the  square  root  of  a  trinomial  square,  extract  the 

square  root  of  the  two  terms  which  are  squares,  connect  them 
with  the  sign  of  the  other  term  of  the  trinomial,  and  prefix  the 
Sign  +  to  the  binomial  thus  formed. 

Example :     Find  the  square  root  of  25x2  _j_  j  gy2  _  40xy . 


V25x2+16y2-40xy=+(V25x2-  V16y2). 
=  +(5x-4y). 

Exercise  76 
Find  the  square  root  of: 

1.  9x2-24xy+16y2 

2.  9+6x+x2 

3.  49m2+14mn+n2 

4.  t2-10tu+25u2 

5.  a^^-2aV+y^^ 

6.  4a6-4a3b2c+bV 

7.  4a2-20ay+25y2 

8.  9m2+42mx+49x2 

9.  -72xy+81x2H-16y2 
10.  25x64-49a^b2-70a2bx3 


112  SQUARES  AND  SQUARE  ROOTS 

Find  the  value  of: 


11.     V30m+25+9m2 


12.  V  -60m2nV+25m4n4+36p8 

13.  V  49a4x2+ 1 12a3x3+64a2x* 


14.      Vx2+x-hi 


15.  yJisi^-\-25h^-\^sih 

16.  VaV'+Vtu+Qu^ 


17.  Vf|m^+2m2n24-||n* 

18.  Vx^-fx^+yfo 

19.  Via%2_|.a2bc3H-2^3— c« 

20.  V|fx«+iV'z'-ixVz 

Square  Root  of  Numbers 
i^5  By  problem  31,  Exercise  73. 

322=  (30+2)2  =  900+1204-4  =  1024. 
.•.V1024=  V  900+ 120+4  =+(30+2)  =  +32. 

To  extract  the  square  root  of  such  numbers  as  1024,  it  is 
necessary  to  separate  them  into  the  form  of  a  trinomial  square. 
•This  can  not  be  done  by  inspection.  Therefore  it  is  convenient 
to  use  the  simplest  form  of  trinomial  square,  t2+2tu+u2,  as 
a  formula.  In  that  case,  t2+2tu+u2  corresponds  to  1024, 
and  its  square  root,  t+u,  corresponds  to  the  square  root  of 
1024,  or  32.     The  work  may  be  arranged  as  follows: 

t+u 
t2+2tu+u2  =  t2+u(2t+u)=    1024       1 30+2 


t2  = 

900 

2t  =  60 
u=  2 

2t+u  =  62 

124  =  u(2t+u) 
124 

".V  1024  =+32. 


SQUARE   ROOT   OF   NUMBERS 


113 


Example  1 :    Find  the  square  root  of  5625. 

In  order  to  find  how  many  digits  there  are  in  the  square  root 
of  a  number,  observe  the  following: 

92  =  81. 
992  =  9801. 
9992  =  998001. 

The  square  of  a  number  of  one  digit  can  not  contain  more 
than  two  digits,  the  square  of  a  number  of  two  digits  can  not 
contain  more  than  four  digits,  etc.  Therefore,  the  number  of 
digits  in  the  square  root  of  a  number  may  be  determined  by 
separating  the  given  number  into  groups  of  two  digits  each, 
beginning  at  the  decimal  point. 


t2+u(2t+u)  =  56'25          170+5 
t2  =  4900 

2t  =  140 
u=     5 

725  =  u(2t+u) 

2t+u=145 

725 

/.V  5625  =+75. 

Observe  that  t  is  found  by  extracting  the  square  root  of  the 
greatest  square  in  the  first  group,  and  u  is  the  integral  number 
found  by  dividing  the  remainder  by  the  number  equal  to  ^t. 

Example  2:    Find  the  value  of  V  289. 


t2+u(2t+u)  = 
t2  = 


2'89 
100 


t+u 
10+9 


2t  =  20 
u  =  9 

2t+u  =  29 


189  =  u(2t+u) 


2  61 


114 


SQUARES  AND  SQUARE  ROOTS 


t+u 

t2+u(2t+u)  = 

=  2'89 

|10+8 

t2  = 

a  00 

2t  =  20 

189  = 

=  u(2t+u) 

u=   8 

2t+u  =  28 

2  24 

? 

t+u 

t2+u(2t+u)  = 

=  2'89 

|10+7 

P  = 

100 

2t  =  20 

189  = 

=  u(2t+u) 

u=  7 

2t+u  =  27 

189 

.-•V  289  =+17. 
Observe  that  in  finding  u,  it  is  not  always  possible  to  take 
the  largest  integral  number  found  by  dividing  the  remainder 
by  the  number  equal  to  2t. 


Exercise  77 

Extract  the  square  root  of: 

1.     1849                       5.     2916 

8.     4624 

2.    3136                       6.    961 

9.     1521 

3.     576                          7.     256 

10.    4489 

4.     5184 

cample:     Find  the  square  root  of  60516. 

t+u 
t2+u(2t+u)=6'05'16  1200+40 
t2  =  4  00  00 

2t  =  4  00 

u=    40 

2t+u  =  4  40 

2  05  16  =  ur2t+u 
1  76  00 

29  16 


SQUARE   ROOT   OF   NUMBERS 


115 


The  square  root  of  60516  will  contain  three  digits.  The 
first  two  are  found  in  the  usual  way.  The  root  is  evidently 
240+ ?  and  the  amount  that  has  been  subtracted  from  60516 
(40000+17600)  is  240^.  Therefore  240  may  be  considered  a 
new  value  of  t,  and  2916  a  new  value  of  u(2t+u),  in  finding 
the  third  digit  of  the  root.     The  problem  then  becomes: 

t+u 
t2+u(2t+u)  =  6'05'16      1240+6 
t2  =  5  76  00 


2t  =  480 
u=     6 

2t+u  =  486 


29  16  =  u(2t+u) 


29  16 


These  two  operations  may  be  combined  into  one  problem  as 
follows: 

t+u  t+u 

t2+u(2t+u)=6'05'16    1200+40    240+6 
t2  =  4  00  00 


2t  =  400 

2  05  16  =  u(2t+u) 

u=  40 

2t+u  =  440 

1  76  00 

2t  =  480 

29  16  =  u(2t+u) 

u=  6 

2t+u  =  486 

29  16 

.-.  V  60516  =  ±246. 


Find  the  value  of 

1.  V  37636 

2.  yllSUi 

3.  V  54756 


Exercise  78 


4.  V 173889 

5.  V  98596 

6.  V  233289 


7.  V  94249 

8.  V 648025 


9.  V  9778129 
10.  V 1022121 


116 


SQUARES  AND  SQUARE  ROOTS 


119  The  operation  of  extracting  square  root  may  be  abridged  as 
follows: 

Find  the  the  value  of: 


V2, 
t2+u(2t-hu)  = 

t2  = 

35.6225 

r 

t      +  u 
1           5. 

t     + 
+  u 

3 

u 
5 

-2' 
=  1 

3     5' 

.6     2'   2 

=  u(2t+u) 

5 

2t=     20 

u=       5 

2t+u=     25 

1 
1 

3     5  = 
2     5 

2t=  300 

u=       3 

2t+u=   303 

1     0 
9 

6     2  =  u(2t+u) 
0     9 

2t  =  3060 

u=       5 

2t+u  =  3065 

1 
1 

5    3     2 
5     3     2 

5  =  u(2t+u) 
5 

/.  V  235.6225  =±15.35 

NOTE:     In  pointing  off  the  given  number  into  groups  of  two  digits 
each,  begin"  at  the  decimal  point  and  proceed  both  right  and  left. 


Exercise  79 
Find  the  square  root  of: 

1.  2323.24 

2.  .120409 

3.  2.6569 

4.  32.1489 

5.  123.4321 


6.  .07557001 

7.  .00003481 

8.  1621.6729 

9.  1040400 
10.  1624.251204 


SQUARE   ROOT   OF   NUMBERS 


117 


120  If  a  number  is  not  a  perfect  square,  the  operation  may  be 
continued  to  as  many  decimal  places  as  is  desired  by  annexing 
a  sufficient  number  of  ciphers. 

Example:     Find  the  value  correct  to  .001  of: 


V4 

.329< 

94. 

t    +    u 

t 

+  u 

t 

+    u 

8 

t 
2. 

+  u 
0 

0          8  =  2.081 

t2+u(2t+u)=4. 
t2  =  4 

32' 

99' 

=  u(2t4 

40'       00 

2t  =  40 
u=  0 

32  = 
00 

-u) 

2t  =  400 
u=     8 

32 
32 

99  = 
64 

=  u(2t+u) 

2t+u  =  408 

2t  =  4160 
u=       0 

35 
00 

40  =  u(2t+u) 
00 

2t =41600 

u=         8 

2t+u  =  41608 

35 
33 

40        00  =  u(2t+u) 
28        64 

11 


36 


/.  V 4. 32994  =+2. 081 

Observe  that  if  3  decimal  places  in  the  result  are  required, 
it  is  necessary  to  determine  the  digit  in  the  4th  place,  and  if 
it  is  5  or  more,  to  add  1  to  the  digit  in  the  3rd  place. 


118  SQUARES  AND  SQUARE  ROOTS 

Exercise  80 

Find  the  square  root  of  the  following  correct  to  4  decimal 
places : 

1.  15  3.     126 

2.  38  4.     2.5 

5.     634.125 

Find  the  value  of  the  following  correct  to  .0001: 

6.  V2  8.      V5 

7.  V3  9.      V.5 

10.      V14.4 
V36=  v"9^=  VV- V4  =  3-2  =  6. 
121  From  this  it  is  evident  that : 

The  square  root  of  a  number  is  equal  to  the  product  of  the  square 
roots  of  its  factors. 

This  law  may  be  used  to  simplify  the  process  of  finding  the 
square  roots  of  numbers  which  contain  one  or  more  factors 
that  are  squares.     For  example: 

VT2  =  vT  V3  =  2  V3  =  +3.4642. 

Exercise  81 

Given  V2^=  1.4142,  V^=  1.7321,  v'5  =  2.2361, 
Find  the  value  of  the  following  correct  to  .001 : 

1.  V~8  6.  V45 

2.  VTS  7.  V48 

3.  V20  8.  V50 

4.  V~27  9.  V72 

5.  V32  10.  V108 


SQUARE  ROOT  OF  FRACTIONS  119 


/'2\2_2  2_£ 
\3J    —  3*3 


11. 

V180 

16. 

V   98 

12. 

V   80 

17. 

V147 

13. 

V125 

18. 

V320 

u. 

V363 

19. 

V243 

16. 

V512 

20. 

V128 

4 

.-.  vl= 

■X3- 

The  square  root  of  a  fraction  is  found  by  extracting  the 
square  root  of  the  numerator  and  of  the  denominator. 


Exercise  82 
Find  the  square  root  of: 
lie  q      -i- 


2. 


"25                                                ^-  64 

3.6.                                               A  121 

4  9                                                *•  9 

O-  2  2^ 


Find  the  value  of  the  following  correct  to  .001: 


6.      Vif  8.      VsV 


7.        V  8T  ^'        ^144 


in        ./  1  62 
10-        V  2  0  0 

123  In  fractions  where  the  denominator  is  not  a  perfect  square 
the  operation  of  finding  the  square  root  may  be  simplified  by 
multiplying  both  numerator  and  denominator  by  a  number 
which  will  make  the  denominator  a  square. 

Example:     vl=  v"A=-^  =  ±?^^^=+.6124 

V16  4  — 


120  SQUARES  AND  SQUARE  ROOTS 

Exercise  83 
Find  the  value  of  the  following  correct  to  .001 : 


1. 

v§ 

2. 

v4 

3. 

vi 

4. 

vi 

6. 

VI 

6. 

VI 

7. 

vf 

8. 

vf 

9. 

VI 

10. 

vV 

Quadratic  Equations 

124  Quadratic  Equation:  A  quadratic  equation  is  one  which  con- 
tains the  square  of  the  unknown  quantity  as  the  highest  power 
of  the  unknown. 

X     13     3x     40 

Example:         -  —  — = — 

2    3x      2      3x 

3x2-26  =  9x2-80  Why? 

54  =  6x2  Why? 

x2  =  9  Why? 

X=  +3  (extracting  the  square  root  of  both 
members) 

Observe  that: 

I.  A  quadratic  equation  of  the  form  x2  =  9  may  be  trans- 
formed into  one  containing  the  first  power  of  the  unknown  by 
extracting  the  square  root  of  both  members. 

II.  In  extracting  the  square  root  of  both  members  of  the 
equation  x2  =  9,  the  full  result  would  be  +x=+3,  which  is  a 
condensed  form  of: 

1.  +x=+3  3.     -x=+3 

2.  +x=-3  4.     -x=-3 

1  and  4,  2  and  3  are  the  same  equations  and  therefore  x=  +3 
expresses  all  four  equations. 


QUADRATIC  EQUATIONS  121 

Exercise  84 
Solve  (correct  to  .001  where  necessary): 

1.  x2=12  7.     x2  =  f 

2.  x2  =  75  8.     x2+10  =  59 

9.     y2-ll  =  185 

10.  7m2- 175  =  0 

11.  8s2-38  =  90 

12.  lla2-5  =  2+2a2 

13.  3(x-2)-x  =  2x(l-x) 

14.  (2t+3)(t+2)-(t+3)(t+4)=4t2-21 

15.  (t-|-4)2+(t-4)2=48 


^^      3xy-l     5(x^-l)     (4x^+1)  _^ 
^^'    —5  lO  25^"" 

y-l         y+1 

5r-3^_r+2 
^***     9r+l     2r+5 

2x-5     ,   -     3x+10 
19.    — TT-  =1.5  — 


3. 

X2  = 

=  55225 

4. 

X2  = 

=  46 

6. 

X2 

169 
1225 

6. 

X2  = 

_  75 

108 

20. 


3  2x+  5 

3x-l     3x+1^29 
3x+l"^3x-l     14 


122  SQUARES    AND    SQUARE    ROOTS 

21.  The  length  of  a  rectangle  is  3  times  its  width,  and  the 
area  is  243  sq.  in.     Find  the  dimensions  of  the  rectangle. 

22.  How  long  must  the  side  of  a  square  field  be  that  the 
area  of  the  field  may  be  5  acres? 

23.  The  dimensions  of  a  rectangle  are  in  the  ratio  f ,  and 
its  area  is  300.     Find  the  dimensions  of  the  rectangle. 

24.  The  side  of  one  square  is  3  times  that  of  another,  and 
its  area  is  96  sq.  in.  more  than  that  of  the  other.  Find  the 
sides  of  the  two  squares. 

25.  If  the  area  of  a  3"  circle  is  28.2744,  find  the  diameter 
of  a  circle  whose  area  is  78.54.     (See  Exercise  67,  problem  3.) 

26.  Find  the  diameter  of  a  circular  piece  of  copper  whose 
weight  is  3.01  oz.  if  a  10"  disk  weighs  9.03  oz.  (See  Exercise 
67,  problem  10.) 

•  27.  The  intensity  of  light  varies  inversely  as  the  square  of 
the  distance  from  the  source  of  light.  How  far  from  a  lamp 
should  a  person  sit  in  order  to  receive  one  half  as  much  light 
as  he  receives  when  sitting  3  ft.  from  the  lamp? 

28.  The  distance  covered  by  a  falling  body  varies  directly 
as  the  square  of  the  time  of  falling.  If  a  ball  drops  402  ft.  in 
5  seconds,  how  long  will  it  take  it  to  drop  600  ft.? 

29.  The  weight  of  an  object  varies  inversely  as  the  square 
of  the  distance  from  the  center  of  the  earth.  If  an  object 
weighs  180  lbs.  at  the  earth's  surface,  at  what  distance  from 
the  center  will  it  weigh  160  lbs.,  if  the  radius  of  the  earth  is 
4000  miles? 

30.  The  surface  of  a  sphere  varies  directly  as  the  square  of 
the  diameter.  Find  the  diameter  of  a  sphere  whose  surface 
is  78.54  sq.  in.,  if  the  surface  of  an  11".  sphere  is  380.1336  sq.  in. 


.     CHAPTER  VIII 

FORMULAS 

Evaluation  of  Formulas  Containing  Square  Root 

Exercise  85 

Evaluate  the  following  formulas  for  the  values  given  (correct 
to  .001  where  necessary): 

1.     h  =  ^V3  when  a  =  5. 


2.  c=Va2+b2  whena  =  4,  b  =  5. 

3.  V  =  2  7r2r2R  when  r  =  J,  R=  l|. 

4.  A  =  |r2V3  whenr  =  3|. 

5.  V  =  ^V2  when  a  =  3.2. 

6.  G=Vab  whena=4,  b  =  5. 

7.  Y  =  — ^ --^-^TTSi^  when  a  =  6,  r  =  18,  R  =  24. 


V  g 


8.  t^TTA/-  when  1=1,  g  =  32. 

9.  s=^(Ar5-l)  whenr  =  2j. 


10.  D=Va2+b2+c2  whena  =  5,  b  =  6,  c  =  7. 

11.  b=  Va2+c2-2a'c  when  a  =14,  a' =  5,  c  =  12. 

12.  l  =  2VD2+a2  +  ttD  when  D  =  16,  a  =  35. 

13.  M  =  |V2(a2+c2)-b2  whena=15,  c  =  17,  b  =  19. 

—  b+Vb2+4ac  ,  o  u     c         on 

14.  x  = when  a  =  3,  b  =  5,  c  =  20. 


2a 


123 


124  FORMULAS 


16.     x= — : >.:cn  a  =  3,  b  =  5,  c  =  20. 

16.  l  =  4/(^^)Va2+7r(5±^)when  D  =  36,  d  =  6,a  =  96. 

17.  s  =  -rV10-2V5  when  r  =  4l. 


N     ^„    C 


18.     s  =  5^.;rR2-yV4R2-0       whenN  =  72,R=  10,C=  13. 


19.    x=  Vr(2r- V4r2-s2)  whenr  =  3,s  =  2. 


20.     A=  Vs(s-a)  (s-b)  (s-c)      when  a=  15,  b=  18,  c  =  22, 

s  =  |(a+b+c). 

125  A  formula  is  an  equation  and  may  be  solved  for  any  of  the 
letters  involved  if  the  values  of  all  the  other  letters  are  given. 

Example  1:     Z  =  4;rra.     Find  a,  when  Z  =  502.656,  r  =  8. 
502.656  =  4.3.1416.8-a 

5. 

15.708 
62.832 
502.656 


a  = 


4-3.1416-8 

Example  2:     V  =  Jrfa.     Find  r,  when  V  =  593.7624,  a  =  7. 

1.0472 
593.7624  =  f  3. 1416- r2. 7 

81 
84.8232 
„    593.7624 


1.04727 

r=+9. 


=  81 


FORMULAS   INVOLVING   SQUARE   ROOT 


125 


Example  3:    b2  =  a2-f  c^— 2ac'.     Solve  for  c',  when  a  =  5, 
b  =  6,  c  =  7. 

36  =  25+49 -2- 5c' 
10c' =  38 
c'  =  3.8 

Exercise  86 

Find  the  values  (correct  to  .001  when  necessary) : 


P=4a 
P=:a+b+c. 


|bh. 


P=2(a-l-b) 
A  =  ab. 
A 

6.  A=Jh(b+b'). 

7.  A=ih(b+bO. 

8.  C  =  27rr. 

9.  A  =  7rr^. 


10. 


1 
w  =  £.p. 


11.     W  =  r.p. 


12.  W  =  r.p. 

13.  L=lfd+J. 

14.  S  =  |gt2. 

16.  S  =  |gt2+vt. 


Find  a,  when  P  =  5§. 

Find  c,  when  P  =  7962,  a  =1728, 
b  =  3154. 

Find  a,  when  P  =  17|,  b  =  2j. 

Find  b,  when  A  =  2.31,  a  =  l.l. 

Find  h,  when  A  =  3U,  b  =  3|. 

Find  h,  when  A  =  96,  b  =  18,  b'  =  6. 

Find  b',  when  A  =  12.8,  b=  1.2,  h  =  8. 

Findr,  when  C  =  50. 

Find  r,  when  A  =  50. 

Find  p,  when  w  =  333 J,  1  =  25,  h  =  4 J. 
Find  1,  when  w=320,h  =  24,p  =  213|. 

Find  h,  when  w  =  150, 1  =  162,  p  =  100. 

Find  d,  when  L=4:yq' 
Find  t,  when  S  =  196.98. 
(See  Exercise  17,  problem  5.) 
Find  V,  when  S  =  164.72,  t  =  3. 


126  FORMULAS 

IIV 

16.  F=— -—.  FmdF,whenu=ll,v  =  7. 

17.  F=-^.  Findv,whenF=liu  =  3. 

u+v     '  ^' 

18.  x  = — ' .  Fmdx,  whenb=-5,  a  =  3,  c=-2. 

19.  A=-r-.  Find  D,  when  A  =115. 

4  ' 

20.  V  =  !rr2a.  Find  r,  if  V  =  330,  a  =  7. 

21.  V  =  ^2a.  Find  a,  if  V  =  46.9,  r  =  2.3. 

22.  V  =  3nr2.?^i5.  ^'^^    ^'   w^^^   V=1932,    H  =  14.6, 

23.  V  =  ;rr2.^^±5.  FindH,  when  V  =  2246,  r  =  8,  h  =  6. 

24.  A  =  ^.  Find  A,  when  a  =  2.3,  b  =  3.2,  c  =  4.L 

^  r  =  2.058. 

25.  A  =  ^-.  Find  r,  when  a  =  21,  b  =  28,  c  =  35, 

A  =  294. 

26.  A  =  |r(a+b+c).     Find  r,  when  a  =  79.3,  b  =  94.2,  c  = 

66.9,A  =  261.012. 

27.  A  =  |r(a+b+c).     Find  a,  when  A  =  27.714,  r  =  2.3095, 

b  =  8,  c  =  8. 

28.  A  =  J(23rR+2zrr)s.  Find  A,  when  R  =  8,  r  =  3,  s  =  7. 

29.  A  =  J(2?rR+23rr)s.  Find  r,  when  A  =  439.824,  R  =  10, 

s=10. 

30.  A=|(2jrR-f-2;rr)s.  Find  s,when A  =  106.029,  R  =  7j,r  =  6- 

31.  l  =  2/(5±^)^+a^  +  .(^). 

Find  1,  when  D  =  l|,  d  =  1  J,  a  =  15. 

32.  h  =  ^V3r  Find  a,  when  h  =  27.7136. 


FORMULAS   INVOLVING   SQUARE   ROOT  127 

33.  A  =  ^  V"3l  Find  h,  when  A  =  y  vl". 

34.  A  =  ^  V"3^  Find  a,  when  A  =  VlS. 

35.  v  =  :^V2^  Find  V,  when  a  =  6. 

Qt.2     

36.  A  =  ^V3.  Find  r,  when  A  =  153. 

37.  c2  =  a2+b2.  Findb,  whenc  =  2.1,  a  =  1.7. 

38.  b2  =  a2+c2+2a'c.  Find  a,  whenb  =  8,  c  =  5,  a'  =  2.1. 

39.  b2  =  a2+c2+2a'c.  Finda',  whena=18,  b  =  16,  c  =  31. 

40.  b2  =  a2+c2-2ac'.  Find  c,  when  a  =  5,  b  =  4,  c'  =  2.3. 

41.  b2  =  a2+c2-2ac'.  Find  c',  when  a=  14,  b  =  15,  c  =  16. 


42.  H  =  r-Vs(s-a)(s-b)(s-c). 

■  FindH,whena  =  2.18,b  =  5,c  =  3.24, 

s  =  J(a+b+c). 

43.  a2+b2  =  2Qy+2m2.   Find  m,  when  a  =  9,  b  =  12,  c  =  15- 

44.  a-+b2  =  2r|y+2m2.    Findb,  whena  =  5,  c  =  13,  m  =  6i. 
46.     s  =  I(vT-l).  Find r,  when 8=10.50685. 


46.  x=     ^+^^'+^^^-    Findx,whena  =  3,b=-7,c=+2. 

^a 

47.  V  =  27r2r2R.  Find  r,  when  V  =  98696.5056,  R  =  50-. 


48.     X  =  V  r (2r  —  V  41^ — s^) .   Find  x,  when  r  =  s  =  10. 

N 


49.     s  =  ^.;rr2-- V4r2-c2. 

Find  N,  when  s  =  23. 1872,  c  =  r=  16. 


— b—  Vb2+4ac  -o-   j       ,  ^  , 

50.     x  = \- — ' .  Findx,when  a  =  — 6,b=  — 9,c=+2. 

^a 


128 


FORMULAS 


Right  Triangle 

126  One  of  the  formulas  most  commonly  used  is  that  of  the  right 
triangle. 


^jb^^C 


Fig.  70 


127  Right  Triangle:  A  right  triangle  is  a  triangle  in  which  one 
angle  is  a  right  angle.  The  lines  including  the  right  angle  are 
called  the  sides,  and  the  line  opposite  the  right  angle  is  called 
the  hypotenuse. 

It  can  be  proved  that; 

128  The  square  of  the  hypotenuse  is  equxil  to  the  sum  of  the  squxires 
of  the  two  sides. 


THE   RIGHT  TRIANGLE 


129 


This  truth  is  stated  by  the  formula: 
c2  =  a2+b2  (Fig.  70). 


Exercise  87 

Find  results  correct  to  .001  when  necessary: 

1.  Find  c,  when  a  =  8,  b  =  15. 

2.  Find  a,  when  b  =  9,  c=41. 

3.  Find  b,  when  a  =  3,  c  =  6. 

4.  Find  the  hypotenuse  of  a  right  triangle  when  the  sides 
are  3.2  and  2.4. 

6.    The  hypotenuse  and  one  side  of  a  right  triangle  are 
respectively  2f  and  l|.     Find  the  other  side. 

6.  The  sides  of  a  right  triangle  are  5j  and  12.5.   Find  the 
hypotenuse. 

7.  The  two  sides  of  a  right  triangle  are  equal  to  each 
other,  and  the  hypotenuse  is  18.     Find  the  sides.     (Fig.  71.) 


\ 

^ 

7 

\^ 

X 

/32 

/tf" 

Fig.  71 

Fig.  72 

Fig.  73 

8.  One  side  of  a  right  triangle  is  3  times  the  other,  and 
the  hypotenuse  is  80.     Find  the  sides.     Draw  a  figure. 

9.  The  two  sides  of  a  right  triangle  are  in  the  ratio  f , 
and  the  hypotenuse  is  225.     Find  the  sides.     Draw  a  figure. 

10.     Find  the  diagonal  of  a  square  whose  sides  are  1.32. 
(Fig.  72.) 


130  FORMULAS 

11.  Find  the  perimeter  of  a  square  whose  diagonal  is  17. 
Draw  a  figure. 

12.  Find  the  diagonal  of  a  rectangle  whose  dimensions  are 
11  and  16.     (Fig.  73.) 

13.  Find  the  dimensions  of  a  rectangle  whose  diagonal  is 
91,  if  the  length  is  5  times  the  width.     Draw  a  figure. 

14.  The  perimeter  of  a  rectangle  is  70,  and  its  sides  are  in 
the  ratio  f.     Find  the  diagonal. 

15.  A  ladder  36  ft.  long  is  placed  with  its  foot  11  ft.  from 
the  base  of  the  building.  How  high  is  a  window  which  the 
ladder  just  reaches? 

16.  A  flag  staff  79  ft.  long  is  broken  29  ft.  from  the  ground. 
If  the  parts  hold  together,  how  far  from  the  foot  of  the  staff 
will  the  top  touch  the  ground? 

17.  How  long  is  a  guy  wire  which  is  attached  to  a  wireless 
tower  227  ft.  from  the  ground,  and  is  anchored  362  ft.  from 
the  foot  of  the  tower? 

18.  The  slant  height  of  a  cone  is  12",  and  the  radius  of  the 
base  is  5|".     Find  the  altitude  of  the  cone.     (Fig.  74.) 


19.  One  side  of  the  base  of  a  square  pyramid  is  14",  and 
the  altitude  is  16".  Find  the  edge,  E.  (Fig.  75.)  (Sugges- 
tion :  The  altitude  of  the  pyramid  meets  the  base  at  the  middle 
point  of  the  diagonal.) 

20.  Find  the  slant  height,  S.     (Fig.  75.) 


INDEX 


SUBJECT  PAGE 

Addition,     Algebraic,     Defini- 
tion         40 

Addition,  Algebraic,  Rule .-     41 

Addition,  Algebraic,  of  several 

numbers    42 

Algebraic  Subtraction,  Defini- 
tion of  45 

Algebraic  Subtraction  Rule.—     4(5 

Angle,  Definition  of 25 

Angle,  Right  25 

Angle,  Straight 25 

Angles,    Complementary 35,  36 

Angles,  Drawing  of 26,  27 

Angles,  Measuring 27,  2S 

Angles,    Reading 28 

Angles,   Sum   of 30,  33 

Angles,  Supplementary 33,  34 

Antecedent    79 

Arm    , 51 

Base  16 

Binomial,  Definition 44 

Binomial,  Square  of 107 

Brace   49 

Bracket   49 

Checking   Equations 24 

Clearing  of  Fractions 9 

Clockwise    — 52 

Coefficient   15 

Coefficient,    Numerical 16 

Complement    35 

Consequent    79 

Counter    Clockwise 52 

Counter   Shaft 99 

Decimals,  Ratios  as 81 

Decimal    Equivalents 81 

Degrees    26 

Division    Law    of    Exponents 

for 68 

Division  Law  of  Sign  for 68 

Division  of  Monomials... .68,  69.  70 


SUBJECT  PAGE 

Division    of    Polynomials    by 

Monomials    '.  72 

Division  of  Polynomials  by 
Polynomials    73,  75 

Equations,  Definition  of 1 

Equation,  Principles  of 10 

Equation,    Checking 24 

Equations,  Quadratic  Defini- 
tion   120 

Equations,  Quadratic,  Solu- 
tion   of 121,  122 

Formula,   Definition '. 19 

Formulas,  Area 22 

Formulas,  Circle 23 

Formulas,  Circular  Ring 23 

Formulas,  Evaluation  of 19 

Formulas,    General 23 

Formulas,     Involving     Square 

Root  123,  124,  125,  126,  127 

Formulas,    Perimeter 20 

Fractions,  Clearing  of 9 

Fulcrum    51 

Gears,  Size  and  R.P.M.  of 103 

Hypotenuse  128 

Law  of  Exponents  for  Divis- 
ion      68 

Law  of  Exponents  for  Multi- 
plication      54 

Law  of  Leverages 57 

Law  of  Signs  for  Division 68 

Law  of  Signs  for  Multiplica- 
tion    53 

Lineshaft    99 

Lever    51 

Leverage   51 

Means  of  a  Proportion.... 86 

Monomial,  Definition 44 

Multiplication 51 


132 


INDEX 


SUBJECT  PAGE 

Multiplication  of  a  Poly- 
nomial by  a  Monomial GO 

Multiplication     of      a     Poly- 
nomial by  a  Polynomial— .62,  63 
Multiplication    of    Monomials 

: 54,  59 

Multiplication  Law  of  Expon- 
ents for 54 

Multiplication   Law   of   Signs 

for    53 

Multiplication    Sign 15 

Negative    Numbers 40 

Members,  Definite 15 

Members,  General  15 

Numbers,  Definite 15 

Numbers,  General 15 

Numbers,  Positive  and  Nega- 
tive     38,  39,  40 

Numbers,  Signed 40 

Order  of  Terms 6 

Parenthesis  16 

Parenthesis,  Kemoval  of 49 

Percentage    81 

Perigon      25 

Perimeter,  Definition 19 

Perimeter,    Formulas 20 

Perimeters,  Equations  involv- 
ing      21 

Polynomial,  Definition 44 

Polynomials,  Addition   of 44 

Positive  Numbers 40 

Power    16 

Proportion,   Definition 86 

Proportion,    Direct 91 

Proportion,  Extremes  of 96 

Proportion,  Inverse 92,  93 

Proportion,  Means  of 86 

Protractor    26 

Pulleys,  R.P.M.  and  Size  of....  99 

Pulleys,  Step,   Cone 101 

Quadratic  Equation,  Defini- 
tion      120 

Quadratic  Equations,  Solu- 
tion   of 121,  122 

Ratio,  Definition 79 

Ration,  Separating  in  a  given  84 


SUBJECT  PAGE 

Ratio,   Terms  of 79 

Ratios,    To    express    as    Deci- 
mals     81 

Right  Triangle,  Definition 128 

Right   Triangle,   Formula 129 

Right  Triangle,  Hypotenuse  of  128 

Right  Triangle,   Sides   of 128 

Rim    Speed 96 

Separating  in  a  Given  Ratio..     84 

Sign   6f   Multiplication 15 

Signed    Numbers 40 

Signs,  Law  of  Signs  for  Divis- 
ion         68 

Signs,  Law  of  Signs  for  Mul- 
tiplication       53 

Signs  of  Grouping 16,  49 

Similar  Terms 5 

Similar    Terms,    Combination 

of    43 

Singular  Terms,  Definition 43 

Specific    Gravity 83 

Speed   i 96 

Speed,    Cutting 97 

Speed,    Rim    or    Surface 96 

Bpeed  Rule 96 

Square  of  a  Binomial 107 

Square   Root,   Definition 109 

Square    Root    of    a    Negative 

No 109 

Square  Root  of  Fractions 119 

Square  Root  of  Monomials 109 

Square  Root  of  Numbers.. 112,  113 
Square  Root  of  Numbers  not 

Perfect    Squares ....117,  118 

Square  Root  Trinomials 110 

Square,    Trinomial 110 

Subtraction,  Algebraic,  Defini- 
tion         45 

Subtraction,  Algebraic,  Rule..     46 
Supplement    33 

Terms,    Definition 43 

Terms,   of   Ratios 79 

Terms,   Order  of 6 

Trinomial,  Definition 44 

Trinomial    Square 110 

Trinomials,  Square  Root  of....  Ill 

Variation    91 

Vinculium  49 


iv;270571 

0-/A  3  9 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


